Chapter 3 Sets (Concepts)

Introduction to Sets – Definition

In mathematics, the concept of a Set is fundamental and serves as a cornerstone for building more complex mathematical ideas. It is a primitive notion, meaning we do not define it formally using simpler terms, but rather describe it based on our intuitive understanding of grouping objects. Informally, a set is a collection of objects. However, for a collection to be considered a mathematical set, it must adhere to specific criteria, primarily being well-defined.

Definition of a Set

A Set is formally defined as a well-defined collection of distinct objects. Let's break down the crucial terms in this definition:

• Well-defined: This is the most important characteristic of a set. A collection is well-defined if, given any object, we can unequivocally determine whether that object belongs to the collection or not. There should be no ambiguity, personal opinion, or subjective judgment involved in deciding membership. The criterion for inclusion must be clear and objective.

• Distinct Objects: The objects within a set are considered unique. Even if an object is listed multiple times when describing a set, it is considered only once as an element of the set. The identity of the elements matters, not how many times they appear in a list. Also, the order in which elements are listed does not affect the set itself.

Understanding "Well-defined Collection"

To reiterate, a collection is well-defined if there is an unambiguous rule for membership. If the criteria for belonging to the collection are vague or depend on subjective factors, it is not a set in the mathematical sense.

Examples of Well-defined Collections (Sets):

• The collection of all states in India. This is well-defined because the political boundaries and recognition of states are clearly defined by the Constitution of India. For any geographical area, we can definitively say if it is a state of India or not.

• The collection of vowels in the English alphabet. This is well-defined; the vowels are precisely defined as {a, e, i, o, u}.

• The collection of all natural numbers less than 10. This is well-defined; the natural numbers are $\{1, 2, 3, ...\}$, and the condition "less than 10" precisely identifies the elements {1, 2, 3, 4, 5, 6, 7, 8, 9}.

• The collection of all rivers in India. While the exact definition of a river might sometimes be debated (e.g., seasonal rivers, tributaries), in a typical context, this collection is generally considered well-defined based on geographical surveys and common understanding.

• The collection of solutions to the equation $x^2 - 4 = 0$. This is well-defined; the solutions are $x=2$ and $x=-2$. The set is $\{2, -2\}$.

Examples of Collections that are NOT Sets:

• The collection of good students in a class. The term "good" is subjective. One person might consider "good" based on high marks, another based on behaviour, punctuality, or participation. There is no single, objective criterion that everyone would agree upon.

• The collection of beautiful flowers in a garden. "Beautiful" is a matter of personal taste and perception, hence subjective and not well-defined.

• The collection of rich people in a city. The term "rich" is relative and lacks a precise, universally agreed-upon definition (e.g., does it mean owning property above a certain value? Having an annual income above a specific amount? These thresholds are arbitrary unless explicitly defined for the context, but generally, the term itself is not mathematically precise).

• The collection of interesting books in a library. "Interesting" is subjective and varies greatly from reader to reader.

The individual items or objects that constitute a set are called its elements or members. Sets are conventionally denoted by capital letters of the English alphabet (such as $A, B, C, X, Y, Z$), while the elements within a set are usually denoted by lowercase letters (like $a, b, c, x, y, z$) or numbers or other mathematical objects.

Membership Notation

To express whether a particular object is an element of a set, we use specific mathematical symbols:

• The symbol $\in$ is used to indicate that an object belongs to or is an element of a set.

  If $x$ is an element of a set $A$, we write it as $x \in A$.

  This is read as "$x$ belongs to $A$" or "$x$ is an element of $A$".

• The symbol $\notin$ is used to indicate that an object does not belong to or is not an element of a set.

  If $y$ is not an element of a set $A$, we write it as $y \notin A$.

  This is read as "$y$ does not belong to $A$" or "$y$ is not an element of $A$".

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Representation of Sets

Once we understand what a set is (a well-defined collection of distinct objects), the next step is to learn how to represent or describe these sets. There are standard methods for writing down sets so that their elements are clearly identified. Primarily, there are two widely used methods for representing a set:

1. The Roster Method or Tabular Form.

2. The Set-Builder Method or Rule Method.

1. Roster Method (or Tabular Form)

In the Roster method, a set is represented by listing all its elements within curly braces `{}`. The elements are separated by commas. This method is also known as the Tabular form because it essentially presents the elements in a list or a table-like structure within the braces.

Key characteristics and conventions of Roster Form:

• Listing Elements: All the elements belonging to the set are explicitly written out.

• Enclosure: The list of elements is enclosed within a pair of curly braces, `{}`.

• Separation: The elements are separated from each other by commas.

• Order Does Not Matter: The order in which the elements are listed within the braces is immaterial. Changing the order of elements does not change the set. For example, the set containing the numbers 1, 2, and 3 can be written as $\{1, 2, 3\}$, $\{1, 3, 2\}$, $\{2, 1, 3\}$, $\{2, 3, 1\}$, $\{3, 1, 2\}$, or $\{3, 2, 1\}$. All these represent the same set.

• Elements Are Distinct (No Repetition): Although the definition of a set specifies distinct objects, when listing elements in Roster form, if an object appears multiple times in the collection from which the set is formed, it is written only once in the set representation. Repetition of elements in the list does not add new elements to the set.

  For example, consider the letters in the word "MATHEMATICS". The collection of letters is M, A, T, H, E, M, A, T, I, C, S. When represented as a set, repeated letters (M, A, T) are listed only once. The set of letters in the word "MATHEMATICS" is $\{M, A, T, H, E, I, C, S\}$. The order does not matter, so $\{A, C, E, H, I, M, S, T\}$ is the same set.

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Sometimes, for sets containing a large number of elements, especially infinite sets where the elements follow a clear pattern, we use an ellipsis (...) to indicate that the pattern continues. However, this is only permissible when the pattern is obvious and easily understandable from the first few listed elements.

• Example: The set of all natural numbers is $\{1, 2, 3, ...\}$. The pattern is adding 1 to the previous number.
• Example: The set of all integers is $\{..., -3, -2, -1, 0, 1, 2, 3, ...\}$. The pattern extends infinitely in both positive and negative directions.
• Example: The set of even integers is $\{..., -4, -2, 0, 2, 4, 6, ...\}$. The pattern is adding 2 to the previous number, extending infinitely.

If the pattern is not clear or the set is finite but very large with no obvious pattern, Roster form may not be practical or possible.

2. Set-Builder Method (or Rule Method)

The Set-Builder method represents a set by stating the characteristic property that all its elements must satisfy, and which no element outside the set satisfies. This method is particularly useful for describing sets that have too many elements to list in Roster form (like infinite sets) or when a specific property defines the set members. It is also known as the Rule method because it provides a rule for membership.

The general form of a set in Set-Builder notation is:

$\{x \mid \text{property } P(x) \text{ is true for } x\}$

Or sometimes a colon is used instead of the vertical bar:

$\{x : \text{property } P(x) \text{ is true for } x\}$

This notation is read as "the set of all elements $x$ such that $x$ has the property $P(x)$".

• The symbol $x$ is a variable representing any generic element of the set.
• The vertical bar ($|$) or colon (:) is read as "such that".
• The statement following the bar or colon is the property that defines the set. Only elements for which this property is true are members of the set.

Key characteristics and conventions of Set-Builder Form:

• Property-Based Definition: The set is defined by a rule or property that its elements satisfy, rather than by listing them individually.

• Conciseness: It provides a concise way to represent large or infinite sets.

• Variable Representation: A variable is used as a placeholder for the elements.

• Requires Understanding of Properties: To understand the set, one must understand the mathematical properties or conditions stated.

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Types of Sets and Their Notations

Sets can be categorized into different types based on the number of elements they contain or based on specific properties of their elements. Understanding these different types and their standard notations is essential for working with sets.

1. Empty Set (or Null Set or Void Set)

A set that contains no elements at all is called the Empty Set. It is also known as the Null Set or the Void Set.

The Empty Set is a unique set, and it is denoted by the symbol $\emptyset$ (pronounced "phi") or by simply writing empty curly braces $\left \{ \right \}$. Both $\emptyset$ and $\left \{ \right \}$ represent the same set.

The cardinality (number of elements) of the empty set is 0, i.e., $n(\emptyset) = 0$.

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Another example: The set of all natural numbers less than 1. Since natural numbers start from 1, there are no natural numbers that are strictly less than 1. So, this set is also the Empty Set.

2. Singleton Set

A set that contains exactly one element is called a Singleton Set.

For example, $\{5\}$ is a singleton set containing the number 5. $\{a\}$ is a singleton set containing the letter 'a'.

The cardinality of a singleton set is 1. If $A = \{x\}$, then $n(A) = 1$.

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3. Finite Set

A set is classified as a Finite Set if it is either the Empty Set or if its elements can be counted completely, meaning that the process of counting the elements comes to an end. The number of elements in such a set is a non-negative integer.

The number of distinct elements in a finite set $A$ is called its cardinality or its order. It is denoted by $n(A)$ or $|A|$.

For example, if $A = \{1, 3, 5, 7\}$, the elements can be counted: there are 4 elements. So, $A$ is a finite set, and its cardinality is $n(A) = 4$.

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4. Infinite Set

A set that is not finite is called an Infinite Set. The elements of an infinite set cannot be counted completely; the process of listing or counting the elements would never come to an end.

Infinite sets can be of different sizes (cardinalities), but that is a concept explored in higher mathematics (countable vs. uncountable infinite sets). For Class 11, understanding that the number of elements is not finite is sufficient.

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5. Equal Sets

Two sets, say $A$ and $B$, are said to be Equal Sets if they contain exactly the same elements. The order in which the elements are listed in Roster form or how many times an element is repeated does not affect the equality of sets.

If sets $A$ and $B$ are equal, we write $A = B$.
If they do not contain exactly the same elements, they are not equal, and we write $A \neq B$.

Equality of sets means that every element of set A is also an element of set B, AND every element of set B is also an element of set A.

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6. Equivalent Sets

Two finite sets $A$ and $B$ are said to be Equivalent Sets if they have the same number of elements. In other words, two finite sets are equivalent if their cardinalities are equal.

We write $A \sim B$ or $A \equiv B$ if A and B are equivalent. This notation signifies that $n(A) = n(B)$.

The concept of equivalence relates to the size of the sets, not the identity of the elements. Two sets are equivalent if their elements can be put into a one-to-one correspondence (a bijection).

Important Note: If two sets are equal, they must have exactly the same elements, and therefore they must have the same number of elements. Thus, **Equal sets are always equivalent**. However, if two sets are equivalent, they only have the same number of elements; the elements themselves might be different. Thus, **Equivalent sets are not necessarily equal**.

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7. Universal Set

In any particular context or problem in set theory, we often deal with elements that come from a larger, overarching set. This fundamental set, which contains all possible elements relevant to a specific discussion or problem, is called the Universal Set.

The Universal Set is typically denoted by the symbol $U$. It is not a fixed set; its definition depends entirely on the context of the problem you are working on. All other sets considered within that specific problem are assumed to be subsets of the Universal Set $U$.

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Other examples of Universal Sets depending on the context:

• If you are discussing sets of natural numbers (e.g., set of even numbers, set of prime numbers), the set of all natural numbers ($\mathbb{N}$) or the set of all integers ($\mathbb{Z}$) or even the set of all real numbers ($\mathbb{R}$) could serve as the Universal Set, depending on how broadly you define the scope of elements you might consider.
• If you are discussing sets of students in a school, the set of all students in that school could be the Universal Set.
• If you are discussing sets of geometrical shapes within a plane, the set of all points in the plane could be the Universal Set.

The choice of the Universal Set is crucial as it defines the boundaries of the elements being considered in a particular problem.

Hierarchy of Number Systems within Real Numbers ($\mathbb{R}$)

When dealing with numbers, the Universal Set is often one of the standard sets of numbers. These sets are nested within each other, forming a hierarchy. The set of Real Numbers ($\mathbb{R}$) can be thought of as the largest of these common number sets used in pre-calculus mathematics.

1. Natural Numbers ($\mathbb{N}$)

These are the counting numbers, the most basic set of numbers.

$\mathbb{N} = \{1, 2, 3, 4, 5, ...\}$

Example Context: If we consider the set of prime numbers $P = \{2, 3, 5, 7, ...\}$, a suitable Universal Set would be $U = \mathbb{N}$.

2. Whole Numbers ($\mathbb{W}$)

This set includes all natural numbers and the number zero.

$\mathbb{W} = \{0, 1, 2, 3, 4, ...\}$

Relation: Every natural number is a whole number. Thus, $\mathbb{N} \subset \mathbb{W}$.

3. Integers ($\mathbb{Z}$)

This set includes all whole numbers and their negative counterparts.

$\mathbb{Z} = \{..., -3, -2, -1, 0, 1, 2, 3, ...\}$

Relation: Every whole number is an integer. Thus, $\mathbb{W} \subset \mathbb{Z}$.

Example Context: To find the solution for the equation $x + 5 = 2$, we get $x = -3$. If our Universal Set was $\mathbb{N}$ or $\mathbb{W}$, this equation would have no solution. A suitable Universal Set would be $U = \mathbb{Z}$.

4. Rational Numbers ($\mathbb{Q}$)

This set includes all numbers that can be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. This includes all integers, terminating decimals, and repeating decimals.

Examples: $\frac{1}{2}, -7, 0.25, 0.\overline{3}$ (which is $\frac{1}{3}$)

Relation: Every integer can be written as a fraction with a denominator of 1 (e.g., $5 = \frac{5}{1}$). Thus, $\mathbb{Z} \subset \mathbb{Q}$.

Example Context: To solve the equation $3x = 7$, we get $x = \frac{7}{3}$. A suitable Universal Set would be $U = \mathbb{Q}$.

5. Irrational Numbers ($\mathbb{I}$ or $\mathbb{Q}'$)

This set includes all real numbers that cannot be expressed as a simple fraction. Their decimal representation is non-terminating and non-repeating.

Examples: $\sqrt{2}, \pi, e$ (Euler's number)

Relation: The set of rational numbers and the set of irrational numbers are disjoint, meaning they have no elements in common. $\mathbb{Q} \cap \mathbb{I} = \emptyset$.

6. Real Numbers ($\mathbb{R}$)

This set is the union of the set of rational numbers and the set of irrational numbers. It includes all numbers that can be represented on the number line.

Relation: $\mathbb{R} = \mathbb{Q} \cup \mathbb{I}$. Both rational and irrational numbers are subsets of real numbers. Thus, $\mathbb{Q} \subset \mathbb{R}$ and $\mathbb{I} \subset \mathbb{R}$.

Example Context: To solve the equation $x^2 = 2$, we get $x = \pm\sqrt{2}$. The solutions are irrational. The appropriate Universal Set for this problem is $U = \mathbb{R}$.

Summary of Relationships and Choice of Universal Set

The hierarchical relationship between these sets can be summarized as:

$\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$

The choice of Universal Set depends entirely on the scope of the problem:

• For problems involving only counting, $U = \mathbb{N}$ is sufficient.
• For problems where negative whole numbers are possible outcomes, $U = \mathbb{Z}$ is necessary.
• For problems involving division or ratios, $U = \mathbb{Q}$ is required.
• For problems in geometry or calculus that involve concepts like square roots of non-perfect squares or $\pi$, $U = \mathbb{R}$ is the standard choice.

Subsets

The concept of a Subset establishes a fundamental relationship between two sets. It helps us describe situations where one collection of objects is entirely included within another collection.

Definition of a Subset

A set $A$ is said to be a Subset of a set $B$ if every element of set $A$ is also an element of set $B$.

In other words, set $A$ is a subset of set $B$ if there is no element in $A$ that is not also in $B$.

This relationship is denoted by the symbol $\subseteq$. We write $A \subseteq B$ to indicate that set $A$ is a subset of set $B$.

In precise mathematical notation, the definition can be stated as:

$A \subseteq B \iff (\forall x, \text{ if } x \in A, \text{ then } x \in B)$

This is read as "Set $A$ is a subset of set $B$ if and only if for all elements $x$, if $x$ belongs to $A$, then $x$ must also belong to $B$".

If set $A$ is not a subset of set $B$, it means that there is at least one element in $A$ which is not present in $B$. We denote this by $A \not\subseteq B$.

$A \not\subseteq B \iff (\exists x \text{ such that } x \in A \text{ and } x \notin B)$

This is read as "Set $A$ is not a subset of set $B$ if and only if there exists at least one element $x$ such that $x$ belongs to $A$ but $x$ does not belong to $B$".

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Proper Subset

A set $A$ is said to be a Proper Subset of a set $B$ if $A$ is a subset of $B$ ($A \subseteq B$) and $A$ is not equal to $B$ ($A \neq B$).

This means that every element of $A$ is in $B$, and there is at least one element in $B$ that is not in $A$. In simpler terms, $A$ is contained within $B$, but $A$ and $B$ are not the same set.

We denote this relationship by $A \subset B$. Note the symbol is $\subset$ (without the underline), indicating strict inclusion.

In mathematical terms:

$A \subset B \iff (A \subseteq B \text{ and } A \neq B)$

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If $A = \{1, 2\}$ and $B = \{1, 2\}$, then $A \subseteq B$ (because every element of A is in B), but $A$ is not a proper subset of $B$ because $A = B$.

Superset

The term Superset is the inverse relationship of a subset. If set $A$ is a subset of set $B$ ($A \subseteq B$), then set $B$ is called a superset of set $A$.

We denote this by $B \supseteq A$, which reads as "$B$ is a superset of $A$".

$B \supseteq A \iff A \subseteq B$

Similarly, if $A$ is a proper subset of $B$ ($A \subset B$), then $B$ is called a Proper Superset of $A$. This means $B$ contains all elements of $A$ plus at least one more element.

We denote this by $B \supset A$, which reads as "$B$ is a proper superset of $A$".

$B \supset A \iff A \subset B$

Using Example 3 again: Since $A = \{1, 2\}$ is a proper subset of $B = \{1, 2, 3\}$, we can also say that $B$ is a proper superset of $A$, i.e., $B \supset A$.

Properties of Subsets

Several important properties arise from the definition of subsets:

• Every set is a subset of itself: For any set $A$, $A \subseteq A$.

  Explanation: By definition, $A \subseteq A$ if every element of $A$ is also an element of $A$. This is trivially true. For any $x$, if $x \in A$, then $x \in A$ is always true.

• The empty set is a subset of every set: For any set $A$, $\emptyset \subseteq A$.

  Explanation: By definition, $\emptyset \subseteq A$ if for every element $x$, if $x \in \emptyset$, then $x \in A$. Since the empty set $\emptyset$ contains no elements, the condition "$x \in \emptyset$" is always false. In logic, an implication "if P, then Q" is true whenever P is false. Thus, "if $x \in \emptyset$, then $x \in A$" is true for any $x$ and any set $A$. This is sometimes called vacuously true.

• Transitive Property of Subsets: If set $A$ is a subset of set $B$ ($A \subseteq B$) and set $B$ is a subset of set $C$ ($B \subseteq C$), then set $A$ is a subset of set $C$ ($A \subseteq C$).

  $(\text{If } A \subseteq B \text{ and } B \subseteq C) \implies A \subseteq C$

  Explanation: Assume $A \subseteq B$ and $B \subseteq C$. Take any element $x \in A$. Since $A \subseteq B$, if $x \in A$, then $x \in B$. Now, since $x \in B$ and $B \subseteq C$, if $x \in B$, then $x \in C$. Combining these, if $x \in A$, then $x \in C$. This is the definition of $A \subseteq C$.

• Set Equality in terms of Subsets: Two sets $A$ and $B$ are equal if and only if $A$ is a subset of $B$ and $B$ is a subset of $A$.

  $A = B \iff (A \subseteq B \text{ and } B \subseteq A)$

  Explanation: This property provides a rigorous way to prove that two sets are equal. It requires showing that every element of A is in B, and every element of B is in A.

Number of Subsets

For a finite set, we can determine exactly how many distinct subsets it has. If a finite set $A$ has $n(A) = m$ elements, the total number of possible subsets of $A$ is given by $2^m$.

$\text{Number of subsets of a set with } m \text{ elements} = 2^m$

Explanation for the formula $2^m$: Consider a set $A = \{a_1, a_2, ..., a_m\}$ with $m$ elements. To form a subset of $A$, for each element $a_i$, we have two choices: either include $a_i$ in the subset or do not include $a_i$. Since there are $m$ elements, and for each element there are 2 independent choices, the total number of ways to make these choices is $2 \times 2 \times ... \times 2$ ($m$ times), which equals $2^m$. Each unique combination of choices forms a unique subset.

The total number of proper subsets of a set $A$ with $n(A) = m$ elements is $2^m - 1$. This is because the only subset that is not a proper subset is the set $A$ itself ($A \subseteq A$ but $A \not\subset A$).

$\text{Number of proper subsets} = 2^m - 1$

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Power Set

The Power Set of a set $A$ is a special set that consists of all possible subsets of $A$. It is often described as the "set of all subsets." This means the elements of the power set are not individual items but are themselves sets.

The power set of $A$ is denoted by $\mathcal{P}(A)$ or $P(A)$.

Cardinality of a Power Set

If $A$ is a finite set with $n(A) = m$ elements, then the number of elements in its power set is equal to the total number of subsets of $A$, which is $2^m$.

$n(\mathcal{P}(A)) = 2^{n(A)} = 2^m$

Derivation of the Formula: To form any subset of $A$, we must decide for each of the $m$ elements whether to include it in the subset or not. For the first element, there are 2 choices (include or exclude). For the second element, there are also 2 choices, and so on for all $m$ elements. Since these choices are independent, the total number of possible combinations of choices is $2 \times 2 \times \dots \times 2$ ($m$ times), which is $2^m$. Each unique combination corresponds to a unique subset. Therefore, there are $2^m$ subsets in total.

Key Characteristics

• The empty set ($\emptyset$) is always a subset of any set $A$, so $\emptyset$ will always be an element of $\mathcal{P}(A)$.
• Any set $A$ is always a subset of itself, so $A$ will always be an element of $\mathcal{P}(A)$.

Important Distinction: Membership vs. Subset

It is crucial to understand the difference between an element belonging to a set ($\in$) and a set being a subset of another set ($\subseteq$).

Consider the set $A = \{1, 2\}$:

• $1 \in A$ (1 is an element of A).
• $\{1\} \subseteq A$ (The set containing 1 is a subset of A).
• $\{1\} \in \mathcal{P}(A)$ (The set containing 1 is an element of the power set of A).
• $1 \notin \mathcal{P}(A)$ (The number 1 itself is not an element of the power set of A; the elements of $\mathcal{P}(A)$ are sets).

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Intervals

In mathematics, particularly when working with real numbers, intervals are special types of subsets of the set of real numbers ($\mathbb{R}$). An interval represents a contiguous range of real numbers between two given endpoints. These endpoints can either be included in the set or excluded, leading to different types of intervals. Intervals are commonly used to express the solution sets of inequalities or to define domains and ranges of functions.

Types of Intervals

Let $a$ and $b$ be two real numbers such that $a < b$. These numbers serve as the endpoints of the interval. Based on whether these endpoints are included or excluded, we define different types of finite intervals:

1. Closed Interval

A Closed Interval consists of all real numbers $x$ that are greater than or equal to the left endpoint $a$ and less than or equal to the right endpoint $b$. Both endpoints, $a$ and $b$, are included in the interval.

Notation: The closed interval is denoted by square brackets enclosing the endpoints: $[a, b]$.

Set-builder form: Using set-builder notation, the closed interval $[a, b]$ is written as $\{x \in \mathbb{R} \mid a \leq x \leq b\}$.

Representation on number line: On a number line, a closed interval is represented by a line segment connecting the endpoints, with solid (filled) dots at $a$ and $b$ to indicate that these points are included in the interval.

2. Open Interval

An Open Interval consists of all real numbers $x$ that are strictly greater than the left endpoint $a$ and strictly less than the right endpoint $b$. The endpoints $a$ and $b$ are NOT included in the interval.

Notation: The open interval is denoted by parentheses enclosing the endpoints: $(a, b)$. Some texts also use reversed square brackets: $]a, b[$. Both notations represent the same open interval.

Set-builder form: Using set-builder notation, the open interval $(a, b)$ is written as $\{x \in \mathbb{R} \mid a < x < b\}$.

Representation on number line: On a number line, an open interval is represented by a line segment connecting the endpoints, with hollow (unfilled) circles or parentheses at $a$ and $b$ to indicate that these points are excluded from the interval.

3. Semi-open or Semi-closed Intervals

These intervals include exactly one of the two endpoints. They are sometimes referred to as half-open or half-closed intervals.

• Left-closed, Right-open Interval:

  This interval consists of all real numbers $x$ such that $x$ is greater than or equal to the left endpoint $a$ and strictly less than the right endpoint $b$. The endpoint $a$ is included, while the endpoint $b$ is not.

  Notation: $[a, b)$.

  Set-builder form: $\{x \in \mathbb{R} \mid a \leq x < b\}$.

  Representation on number line: A line segment with a solid dot at $a$ and a hollow circle (or parenthesis) at $b$.

• Left-open, Right-closed Interval:

  This interval consists of all real numbers $x$ such that $x$ is strictly greater than the left endpoint $a$ and less than or equal to the right endpoint $b$. The endpoint $a$ is not included, while the endpoint $b$ is included.

  Notation: $(a, b]$.

  Set-builder form: $\{x \in \mathbb{R} \mid a < x \leq b\}$.

  Representation on number line: A line segment with a hollow circle (or parenthesis) at $a$ and a solid dot at $b$.

Infinite Intervals

Intervals can also extend infinitely in one or both directions along the real number line. These intervals use the symbols for positive infinity ($\infty$) and negative infinity ($-\infty$). Note that $\infty$ and $-\infty$ are not real numbers, so they are always associated with an open boundary (parenthesis).

• $[a, \infty)$: The set of all real numbers $x$ such that $x$ is greater than or equal to $a$. $\{x \in \mathbb{R} \mid x \geq a\}$.

• $(a, \infty)$: The set of all real numbers $x$ such that $x$ is strictly greater than $a$. $\{x \in \mathbb{R} \mid x > a\}$.

• $(-\infty, b]$: The set of all real numbers $x$ such that $x$ is less than or equal to $b$. $\{x \in \mathbb{R} \mid x \leq b\}$.

• $(-\infty, b)$: The set of all real numbers $x$ such that $x$ is strictly less than $b$. $\{x \in \mathbb{R} \mid x < b\}$.

• $(-\infty, \infty)$: The set of all real numbers. $\{x \in \mathbb{R}\}$. This represents the entire number line, $\mathbb{R}$.

Relationship between Intervals and Inequalities

Interval notation is a convenient shorthand for expressing inequalities involving real numbers.

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The given inequality is $x \geq 10$. This means $x$ can be 10 or any real number greater than 10.

This is an infinite interval starting at 10 and extending to positive infinity.

The endpoint 10 is included (due to $\geq$), so we use a square bracket '[' at 10.

The interval extends to positive infinity ($\infty$), which always uses a parenthesis ')'.

Interval Notation: $[10, \infty)$.

In set-builder form, we describe the property the real numbers $x$ must satisfy:

Set-builder Form: $\{x \in \mathbb{R} \mid x \geq 10\}$.

Representation on number line: Draw a number line. Place a solid dot at 10 and draw an arrow extending from this dot to the right, covering all numbers greater than 10.

Venn Diagrams

When working with sets, especially when dealing with relationships between multiple sets or performing operations on them, visual representation can be extremely helpful. Venn Diagrams are graphical illustrations used to show the relationships between sets and their elements. They provide a simple and intuitive way to understand set theory concepts and solve problems involving set operations. These diagrams were introduced by the English logician and philosopher John Venn in the late 19th century.

Components of a Venn Diagram

A typical Venn diagram consists of the following components:

1. Universal Set (U)

The Universal Set, which represents the entire scope of elements under consideration for a particular problem, is usually depicted by a large rectangle. All other sets discussed in the problem are considered to be subsets of this Universal Set. The area within the rectangle represents all the elements in the Universal Set $U$.

2. Sets (Subsets of U)

Individual sets that are subsets of the Universal Set are commonly represented by circles or ovals drawn within the rectangle. Each circle represents a specific set, and the elements belonging to that set are considered to be inside that circle.

3. Elements

The specific elements of the sets are indicated as points within the corresponding regions. If an element belongs to multiple sets, it is placed in the overlapping region. If an element is in the Universal Set but not in any of the specific sets, it is placed within the rectangle but outside all the circles.

In the diagram above, for $U=\{1,2,3,4,5,6,7,8\}$, $A=\{1,2,3\}$ and $B=\{3,4,5\}$, the elements are placed in their respective regions.

Representing Set Relationships using Venn Diagrams

Venn diagrams are powerful tools for visually depicting the relationships between two or more sets:

1. Disjoint Sets

Two sets $A$ and $B$ are said to be disjoint sets if they have no elements in common. Their intersection is the empty set. In a Venn diagram, disjoint sets are represented by two circles that do not overlap at all.

$A \cap B = \emptyset$

Example: If $A = \{1, 2\}$ and $B = \{3, 4\}$, the circles for A and B would be separate.

2. Intersecting Sets

Two sets $A$ and $B$ are called intersecting sets if they have at least one element in common. In a Venn diagram, this is shown by two overlapping circles. The overlapping region represents the elements common to both sets, i.e., the intersection $A \cap B$.

$A \cap B \neq \emptyset$

Example: If $A = \{1, 2, 3\}$ and $B = \{2, 3, 4\}$, the overlapping region would contain elements 2 and 3.

3. Subset

If set $A$ is a subset of set $B$, every element of $A$ is also an element of $B$. This is shown by drawing the circle for set $A$ completely inside the circle for set $B$.

$A \subseteq B$

Example: If $A = \{1, 2\}$ and $B = \{1, 2, 3\}$, the circle for A would be inside the circle for B.

Representing Set Operations using Venn Diagrams

Venn diagrams are especially useful for visualizing the outcome of set operations. The result of an operation is typically shown by shading the relevant region(s).

1. Union of Sets ($A \cup B$)

The union of sets A and B contains all elements that are in A, or in B, or in both. The shaded region covers both circles entirely.

$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$

2. Intersection of Sets ($A \cap B$)

The intersection of sets A and B contains only the elements that are common to both A and B. The shaded region is the overlapping part of the two circles.

$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

3. Difference of Sets ($A - B$)

The difference A - B contains the elements that are in A but not in B. The shaded region is the part of circle A that does not overlap with circle B.

$A - B = \{x \mid x \in A \text{ and } x \notin B\}$

4. Complement of a Set ($A'$)

The complement of set A contains all elements in the Universal Set U that are not in A. The shaded region is the area inside the rectangle but outside of circle A.

$A' = \{x \in U \mid x \notin A\}$

Operations on Sets

Just as we perform arithmetic operations like addition, subtraction, multiplication, and division on numbers to combine or compare them, we can also perform operations on sets. These set operations allow us to combine sets, find common elements, or find elements present in one set but not another. Understanding these operations is crucial for solving problems involving sets and for building more complex set-theoretic concepts.

Throughout this discussion on set operations, let $U$ denote the Universal Set (the set of all elements under consideration in a given context), and let $A$ and $B$ be any two subsets of $U$.

1. Union of Sets

The Union of two sets $A$ and $B$ is a new set that contains all the elements that are in set $A$ or in set $B$ or in both sets $A$ and $B$. In simpler terms, it is the collection of all elements from both sets combined, with common elements listed only once (as elements in a set must be distinct).

The union of sets $A$ and $B$ is denoted by the symbol $\cup$. We write it as $A \cup B$.

In Set-builder form, the definition of the union of two sets is:

$A \cup B = \{x \mid x \in A \text{ or } x \in B\}$

This is read as "the set of all elements $x$ such that $x$ is an element of $A$ or $x$ is an element of $B$". Note that the word "or" in mathematics is typically used in the inclusive sense, meaning "A or B or both".

Answer:

2. Intersection of Sets

The Intersection of two sets $A$ and $B$ is a new set that contains only those elements that are common to both set $A$ and set $B$.

The intersection of sets $A$ and $B$ is denoted by the symbol $\cap$. We write it as $A \cap B$.

In Set-builder form, the definition of the intersection of two sets is:

$A \cap B = \{x \mid x \in A \text{ and } x \in B\}$

This is read as "the set of all elements $x$ such that $x$ is an element of $A$ and $x$ is an element of $B$".

Answer:

If two sets have no elements in common, their intersection is the Empty Set. Such sets are called Disjoint Sets.

Sets $A$ and $B$ are disjoint $\iff A \cap B = \emptyset$

Example: If $A = \{1, 2\}$ and $B = \{3, 4\}$, then $A \cap B = \emptyset$. Sets A and B are disjoint.

3. Difference of Sets

The Difference of set $A$ and set $B$ (in that order) is the set containing all the elements that are in set $A$ but are not in set $B$.

The difference of set $A$ and set $B$ is denoted by $A - B$ or $A \setminus B$.

In Set-builder form, the definition is:

$A - B = \{x \mid x \in A \text{ and } x \notin B\}$

This is read as "the set of all elements $x$ such that $x$ is an element of $A$ and $x$ is not an element of $B$".

Similarly, the difference of set $B$ and set $A$, denoted by $B - A$, is the set of elements that are in $B$ but not in $A$.

$B - A = \{x \mid x \in B \text{ and } x \notin A\}$

It is important to note that generally, $A - B$ is not equal to $B - A$.

Answer:

4. Complement of a Set

The Complement of a set $A$ is the set containing all the elements in the Universal Set $U$ that are not in set $A$. The complement of $A$ depends on the definition of the Universal Set for that particular problem.

The complement of set $A$ is denoted by $A'$ or $A^c$ or $\overline{A}$.

In Set-builder form, the definition of the complement of a set is:

$A' = \{x \mid x \in U \text{ and } x \notin A\}$

This is read as "the set of all elements $x$ such that $x$ is an element of the Universal Set $U$ and $x$ is not an element of $A$".

From the definition, we can see that the complement of $A$ is equivalent to the difference between the Universal Set and the set $A$:

$A' = U - A$

Answer:

Properties and Laws of Set Operations

Set operations obey various laws, which are analogous to the laws of algebra for numbers. These laws are useful for simplifying expressions involving sets and proving set identities.

• Idempotent Laws: Performing the union or intersection of a set with itself results in the same set.

  $A \cup A = A$

  $A \cap A = A$

• Identity Laws: These involve the empty set ($\emptyset$) and the universal set ($U$).

  $A \cup \emptyset = A$

  $A \cap U = A$

  $A \cap \emptyset = \emptyset$

  $A \cup U = U$

• Commutative Laws: The order of sets in union and intersection operations does not affect the result.

  $A \cup B = B \cup A$

  $A \cap B = B \cap A$

• Associative Laws: When performing union or intersection with three or more sets, the grouping of sets does not affect the result.

  $(A \cup B) \cup C = A \cup (B \cup C)$

  $(A \cap B) \cap C = A \cap (B \cap C)$

• Distributive Laws: These laws show how union and intersection interact with each other.

  $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

  $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$

• De Morgan's Laws: These laws relate the complement of unions and intersections to the complements of the individual sets.

  $(A \cup B)' = A' \cap B'$

  $(A \cap B)' = A' \cup B'$

• Complement Laws: These involve the relationship between a set and its complement with respect to the universal set.

  $A \cup A' = U$

  $A \cap A' = \emptyset$

  $(A')' = A$

  (The complement of the complement of a set is the set itself)

  $\emptyset' = U$

  (The complement of the empty set is the universal set)

  $U' = \emptyset$

  (The complement of the universal set is the empty set)

• Difference in terms of Complement and Intersection: The difference $A-B$ can also be expressed using complement and intersection.

  $A - B = A \cap B'$

  Explanation: $A-B$ consists of elements in A but not in B. $B'$ consists of elements not in B. $A \cap B'$ consists of elements that are in A AND not in B. This matches the definition of $A-B$.

Formulae for Number of Elements (Cardinality)

When dealing with finite sets, we are often interested in the number of elements in the sets resulting from set operations. These formulae are particularly useful in solving practical problems involving surveys or data classification.

Principle of Inclusion-Exclusion for Two Sets

For any two finite sets $A$ and $B$, the number of elements in their union, $n(A \cup B)$, is given by the formula:

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$

Derivation:

If sets $A$ and $B$ are disjoint, meaning $A \cap B = \emptyset$, then the number of elements in their intersection is $n(A \cap B) = 0$. In this special case, the formula for the union simplifies to:

If $A \cap B = \emptyset$, then $n(A \cup B) = n(A) + n(B)$

Formulae for Difference and Complement Cardinality

The number of elements in the difference of two sets can also be expressed in terms of cardinality:

$n(A - B) = n(A) - n(A \cap B)$

Derivation:

Similarly, the number of elements in the complement of a set A with respect to a universal set $U$ is the total number of elements in $U$ minus the number of elements in A:

$n(A') = n(U) - n(A)$

Derivation:

The union of two sets can also be expressed as the sum of the number of elements in $A$ only, $B$ only, and the number of elements in their intersection:

$n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)$

Explanation: The sets $(A-B)$, $(B-A)$, and $(A \cap B)$ are mutually disjoint regions that together form $A \cup B$. Summing their cardinalities gives the total cardinality of the union.

Principle of Inclusion-Exclusion for Three Sets

For three finite sets $A$, $B$, and $C$, the number of elements in their union is given by:

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C)$

Explanation: When we sum $n(A) + n(B) + n(C)$, elements in pairwise intersections are counted twice. We subtract $n(A \cap B)$, $n(B \cap C)$, and $n(A \cap C)$ to correct this. However, the elements in the intersection of all three sets ($A \cap B \cap C$) were counted three times initially and then subtracted three times, resulting in a count of zero for that region. Therefore, we must add back $n(A \cap B \cap C)$ once to get the correct total.

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