Sets: Types and Cardinality

Types of Sets (Empty, Finite, Infinite, Singleton, Equal, Equivalent)

Sets can be classified into various types based on the number of elements they contain, their comparison with other sets, and other properties. Understanding these types is essential for working with sets effectively.

1. Empty Set (Null Set or Void Set)

An empty set is a set that contains no elements at all. It is the unique set with zero elements.

Notation:

The empty set is denoted by the symbol $\emptyset$ (read as "phi" or "the empty set") or by a pair of empty curly braces, $\{\}$.

Important Considerations:

• The set $\{\emptyset\}$ is not the empty set. It is a set that contains one element, and that element is the empty set itself. Thus, it is a singleton set (see below) with cardinality 1.
• The set $\{0\}$ is not the empty set. It is a set containing the element '0'. It is also a singleton set with cardinality 1. The element 0 is different from the concept of an empty set.

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2. Finite Set

A set is called a finite set if it is either the empty set or if its elements can be counted using a process that eventually comes to an end. In other words, a finite set has a definite, limited number of elements.

Examples:

• The set of all days in a week: $D = \{\text{Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}\}$. This set has 7 elements.
• The set of all states in India. This number is definite (currently 28).
• The set of solutions to the equation $x^2 - 5x + 6 = 0$. Factoring, we get $(x-2)(x-3)=0$, so $x=2$ or $x=3$. The set is $\{2, 3\}$, which has 2 elements.
• The set of letters in the word "SCHOOL": $\{S, C, H, O, L\}$. This set has 5 elements.
• The empty set $\emptyset$ is considered a finite set because the number of elements is 0, which is a definite, non-negative integer.

3. Infinite Set

A set that is not finite is called an infinite set. It contains an unlimited number of elements, and the process of counting its elements would never come to an end.

Examples:

• The set of natural numbers: $\mathbb{N} = \{1, 2, 3, ...\}$. There is no last natural number.
• The set of integers: $\mathbb{Z} = \{..., -2, -1, 0, 1, 2, ...\}$. It extends infinitely in both positive and negative directions.
• The set of rational numbers: $\mathbb{Q}$.
• The set of real numbers: $\mathbb{R}$.
• The set of points on a line.
• $\{x \in \mathbb{N} \mid x \text{ is even}\} = \{2, 4, 6, 8, ...\}$.

Distinguishing Finite and Infinite Sets:

Sometimes, a set defined by a property might seem simple but turns out to be infinite. For example, the set of all points on the circumference of a circle is an infinite set.

4. Singleton Set (Unit Set)

A set containing exactly one element is called a singleton set or a unit set.

Examples:

• $\{5\}$ is a singleton set containing the element 5.
• $\{\emptyset\}$ is a singleton set containing the empty set as its only element.
• $\{x \in \mathbb{Z} \mid x^2 = 0\}$. The only integer $x$ satisfying $x^2=0$ is $x=0$. So, this set is $\{0\}$.
• $\{x \mid x \text{ is the current Prime Minister of India}\}$. Assuming there is only one Prime Minister at any given time, this set is a singleton set containing that person.

5. Equal Sets

Two sets $A$ and $B$ are said to be equal if and only if they contain exactly the same elements. If two sets are equal, we write $A = B$. Otherwise, we write $A \neq B$.

Condition for Equality:

Set $A$ is equal to set $B$ ($A = B$) if and only if every element of $A$ is an element of $B$, and every element of $B$ is an element of $A$.

• Symbolically: $A = B \iff ( \forall x, x \in A \iff x \in B )$. This is equivalent to saying that $A$ is a subset of $B$ and $B$ is a subset of $A$ ($A \subseteq B$ and $B \subseteq A$).
• As mentioned before, the order in which elements are listed does not matter, and repetition of elements is ignored when determining the elements of a set. So, $\{1, 2, 3\}$, $\{3, 1, 2\}$, and $\{1, 1, 2, 3, 3\}$ all represent the same set and are equal to each other.

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6. Equivalent Sets

Two finite sets $A$ and $B$ are said to be equivalent if they have the same number of elements. Equivalence is about the size of the sets, not the identity of the elements.

Condition for Equivalence:

Set $A$ is equivalent to set $B$ if their cardinal numbers are equal.

Equivalence is sometimes denoted by the symbol $\sim$ or $\leftrightarrow$. For example, $A \sim B$ or $A \leftrightarrow B$ means set A is equivalent to set B.

Distinction between Equal Sets and Equivalent Sets:

• If two sets are equal, they must contain exactly the same elements. Therefore, they must have the same number of elements. So, equal sets are always equivalent.
• If two sets are equivalent, they only have the same number of elements. The elements themselves might be different. So, equivalent sets are not necessarily equal.

Think of it like two groups of people. If the groups are equal, they contain the exact same individuals. If the groups are equivalent, they just have the same headcount, but the individuals might be different.

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Cardinal Number of a Finite Set

The cardinal number, also known as the cardinality, of a finite set is a measure of the "size" of the set. It represents the count of the distinct elements contained within the set.

Definition:

For a finite set $A$, its cardinal number is the number of distinct elements in $A$.

Notation:

The cardinal number of a set $A$ is commonly denoted by $n(A)$ or $|A|$.

Key Points:

• The cardinal number is always a non-negative integer ($0, 1, 2, 3, ...$).
• The cardinal number of the empty set $\emptyset$ is 0, as it has no elements:

  $n(\emptyset) = |\emptyset| = 0$

• If a set $A$ is a singleton set, it contains exactly one element, so its cardinal number is 1:

  If $A = \{a\}$, then $n(A) = |A| = 1$

• For an infinite set, the cardinality is not a finite number. The concept of cardinality is extended to infinite sets (using transfinite cardinal numbers like $\aleph_0$, $\mathfrak{c}$), but this is typically covered in advanced set theory. For basic set theory, we simply identify whether a set is finite or infinite. If finite, we find its specific cardinal number; if infinite, we state that it is infinite.

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Subsets and Superset

Comparing sets based on their elements gives rise to the concepts of subsets and supersets. These concepts describe relationships where all elements of one set are also found in another set.

Subset ($\subseteq$)

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

Notation:

We denote that $A$ is a subset of $B$ by writing $A \subseteq B$. This notation is read as "$A$ is a subset of $B$", "$A$ is contained in $B$", or "$A$ is included in $B$".

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

Formal Definition:

(For all elements $x$, if $x$ is in $A$, then $x$ is also in $B$).

Key Properties of Subsets:

• Every set is a subset of itself:

  For any set $A$, $A \subseteq A$. This is because every element in $A$ is trivially also in $A$.

• The empty set is a subset of every set:

  For any set $A$, $\emptyset \subseteq A$. This is considered vacuously true. The condition for $\emptyset \subseteq A$ is "for all $x$, if $x \in \emptyset$, then $x \in A$". Since there are no elements $x$ in $\emptyset$, the premise "$x \in \emptyset$" is always false, and an implication with a false premise is always true. Hence, the condition holds for any set $A$.

• Transitivity of Subsets:

  If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$. If every element of $A$ is in $B$, and every element of $B$ is in $C$, then it logically follows that every element of $A$ must be in $C$.

• Equality in terms of Subsets:

  Two sets $A$ and $B$ are equal ($A = B$) if and only if $A$ is a subset of $B$ and $B$ is a subset of $A$.

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

Superset ($\supseteq$)

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

Notation:

We denote that $B$ is a superset of $A$ by writing $B \supseteq A$. This notation is read as "$B$ is a superset of $A$" or "$B$ contains $A$".

Equivalence:

The statement $A \subseteq B$ is completely equivalent to the statement $B \supseteq A$. They express the same relationship between the two sets from different perspectives.

Proper Subset ($\subset$ or $\subsetneq$)

A set $A$ is called 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 all elements of $A$ are in $B$, but $B$ contains at least one element that is not in $A$.

Notation:

There are two common notations for proper subsets:

• $A \subset B$: This notation is widely used, especially in introductory texts. It explicitly means $A \subseteq B$ and $A \neq B$.
• $A \subsetneq B$: This notation is sometimes used to specifically emphasize that $A$ is a proper subset and cannot be equal to $B$.

We will use the notation $A \subset B$ to denote a proper subset.

Formal Definition:

An equivalent way to state the condition for a proper subset is:

(Every element of A is in B, AND there exists at least one element y in B such that y is not in A).

Key Properties of Proper Subsets:

• The empty set $\emptyset$ is a proper subset of every non-empty set. If $B$ is not empty, $\emptyset \subseteq B$ and $\emptyset \neq B$.
• No set is a proper subset of itself ($A \not\subset A$) because $A$ is always equal to $A$.

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Distinction between 'Element of' ($\in$) and 'Subset of' ($\subseteq$)

It is crucial to understand and correctly use the symbols $\in$ and $\subseteq$. They represent fundamentally different relationships:

• The symbol $\in$ relates an element to a set. It signifies that an object is a member of a set.
• The symbol $\subseteq$ relates a set to another set. It signifies that one set is contained within another set.

Example:

Let $S = \{1, \{2, 3\}, 4\}$. This set $S$ has three elements: the number 1, the set $\{2, 3\}$, and the number 4.

• $1 \in S$: Correct. 1 is an element of the set S.
• $\{2, 3\} \in S$: Correct. The set $\{2, 3\}$ is listed as one of the elements of S.
• $4 \in S$: Correct. 4 is an element of the set S.
• $2 \notin S$: Correct. The number 2 itself is not an element directly listed in S.
• $3 \notin S$: Correct. The number 3 itself is not an element directly listed in S.
• $\{1\} \subseteq S$: Correct. The set containing only the element 1 is a subset of S because $1 \in S$.
• $\{\{2, 3\}\} \subseteq S$: Correct. The set whose only element is the set $\{2, 3\}$ is a subset of S, because $\{2, 3\}$ is an element of S.
• $\{1, 4\} \subseteq S$: Correct. The set containing 1 and 4 is a subset of S because both $1 \in S$ and $4 \in S$.
• $\{2\} \not\subseteq S$: Correct. The set containing only 2 is not a subset of S because $2 \notin S$. (For $\{2\}$ to be a subset of S, its element 2 must be in S).
• $\{1, \{2, 3\}\} \subseteq S$: Correct. This set is a subset of S because both $1$ and $\{2, 3\}$ are elements of S.

Understanding this distinction is vital for avoiding common mistakes in set theory problems.

Power Set

The power set of a given set is a collection that brings together all the possible subsets of that set. It's a way of forming a new set whose elements are themselves sets.

Definition of Power Set

The power set of a set $A$ is the set containing all possible subsets of $A$.

Notation:

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

Key Characteristics:

• The elements of the power set $P(A)$ are sets themselves (specifically, they are the subsets of $A$).
• For any set $A$, the empty set $\emptyset$ is always a subset of $A$. Therefore, $\emptyset$ is always an element of $P(A)$, i.e., $\emptyset \in P(A)$.
• For any set $A$, the set $A$ itself is always a subset of $A$ ($A \subseteq A$). Therefore, $A$ is always an element of $P(A)$, i.e., $A \in P(A)$.

Cardinality of the Power Set

If a finite set $A$ has $n$ elements, i.e., its cardinality is $n$ ($|A| = n$), then the number of elements in its power set $P(A)$ is $2$ raised to the power of $n$.

Reasoning for the Cardinality Formula:

Consider a set $A = \{a_1, a_2, ..., a_n\}$ with $n$ elements. To form any subset of $A$, we go through each element of $A$ and decide whether to include it in the subset or not.

• For element $a_1$, we have 2 choices: include it or exclude it.
• For element $a_2$, we have 2 choices: include it or exclude it.
• ...
• For element $a_n$, we have 2 choices: include it or exclude it.

Since there are $n$ elements and 2 independent choices for each element, the total number of ways to make these choices is $2 \times 2 \times ... \times 2$ (repeated $n$ times).

Each distinct combination of these choices forms a unique subset of $A$. Therefore, the total number of subsets is $2^n$, which is the cardinality of the power set.

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Universal Set

When working with sets in a particular context, it is often helpful to define a larger set that encompasses all the elements being discussed or considered in that specific problem or domain. This overarching set is known as the universal set.

Definition of Universal Set

A universal set is a set that contains all the elements relevant to a particular context or discussion. It is the basic set within which all other sets in that context are considered. The universal set is not fixed; its definition depends entirely on the specific problem or area of mathematics being studied.

Notation:

The universal set is commonly denoted by the capital letter $U$. Sometimes, especially in certain fields like probability or statistics, the Greek letter $\xi$ (xi) or $\Omega$ (Omega) might be used.

Relationship to Other Sets:

By definition, any set $A$ being discussed within a specific context is considered a subset of the universal set defined for that context.

The universal set serves as the frame of reference for all other sets in a given problem.

Examples Illustrating the Context-Dependent Nature of the Universal Set:

• Example 1: Discussing Integers

  If we are studying sets of integers, such as the set of even integers $E = \{..., -4, -2, 0, 2, 4, ...\}$, the set of odd integers $O = \{..., -3, -1, 1, 3, ...\}$, or the set of prime numbers $P = \{2, 3, 5, 7, 11, ...\}$, a suitable universal set would be the set of all integers, $\mathbb{Z}$. In this context, $U = \mathbb{Z}$, and $E \subseteq U$, $O \subseteq U$, $P \subseteq U$.

• Example 2: Discussing Students in a School

  If we are analyzing sets of students within a particular school, such as the set of students in Class XII, the set of students who play football, or the set of students who opted for science stream, the universal set $U$ would naturally be the set of all students in that school.

• Example 3: Discussing Geometric Shapes in a Plane

  If we are working with sets of geometric shapes drawn on a piece of paper, such as the set of all triangles, the set of all circles, or the set of all squares, the universal set $U$ could be the set of all points in the plane (or the paper itself).

• Example 4: Discussing Solutions to Equations

  If we are solving equations and considering the solutions, the universal set is often the set of numbers we are allowed to use. For example, if we are looking for integer solutions, $U = \mathbb{Z}$. If we are looking for real solutions, $U = \mathbb{R}$. If we are looking for complex solutions, $U = \mathbb{C}$.

The universal set provides the boundaries for our operations and definitions within a given problem. Its clear definition is particularly important when discussing concepts like the complement of a set.

Intervals as Subsets of Real Numbers

Intervals are a specific type of subset of the set of real numbers ($\mathbb{R}$). They represent a continuous range of real numbers on the number line between two given points, which are called the endpoints of the interval. Intervals are widely used in calculus, inequalities, and analysis.

Types of Finite Intervals

Let $a$ and $b$ be two real numbers such that $a < b$. We can define four types of finite intervals using $a$ and $b$ as endpoints:

1. Open Interval: $(a, b)$

   • Set-Builder Notation: $\{x \in \mathbb{R} \mid a < x < b\}$
   • Description: This interval includes all real numbers that are strictly greater than $a$ and strictly less than $b$. The endpoints $a$ and $b$ are not included in the interval.
   • Number Line Representation: On a number line, this is shown with open circles (or parentheses) at the endpoints $a$ and $b$, indicating that the points themselves are excluded, and a shaded line between them.
     
     (Imagine an image showing a number line with points 'a' and 'b', open circles at 'a' and 'b', and the segment between 'a' and 'b' shaded.)

2. Closed Interval: $[a, b]$

   • Set-Builder Notation: $\{x \in \mathbb{R} \mid a \le x \le b\}$
   • Description: This interval includes all real numbers that are greater than or equal to $a$ and less than or equal to $b$. Both endpoints $a$ and $b$ are included in the interval.
   • Number Line Representation: On a number line, this is shown with closed (filled) circles (or square brackets) at the endpoints $a$ and $b$, indicating that the points themselves are included, and a shaded line between them.
     
     (Imagine an image showing a number line with points 'a' and 'b', closed circles at 'a' and 'b', and the segment between 'a' and 'b' shaded.)

3. Semi-open (or Semi-closed) Intervals:

   • $[a, b)$: Includes $a$ but excludes $b$.

     Set-Builder Notation:

     $\{x \in \mathbb{R} \mid a \le x < b\}$

     Description:

     Real numbers greater than or equal to $a$ and strictly less than $b$. Endpoint $a$ is included, $b$ is excluded.

     Number Line:

     
     (Imagine a number line with a closed circle at 'a', an open circle at 'b', and shading between them.)
   • $(a, b]$: Excludes $a$ but includes $b$.

     Set-Builder Notation:

     $\{x \in \mathbb{R} \mid a < x \le b\}$

     Description:

     Real numbers strictly greater than $a$ and less than or equal to $b$. Endpoint $a$ is excluded, $b$ is included.

     Number Line:

     
     (Imagine a number line with an open circle at 'a', a closed circle at 'b', and shading between them.)

The length of any of these finite intervals $(a, b)$, $[a, b]$, $[a, b)$, or $(a, b]$ is given by the difference between the endpoints, which is $b - a$.

Types of Infinite Intervals

Intervals can also extend infinitely in one or both directions along the real number line. We use the symbols $\infty$ (positive infinity) and $-\infty$ (negative infinity) to represent these unbounded directions. Note that $\infty$ and $-\infty$ are not real numbers themselves, so they are always associated with an open bracket '('.

• $(a, \infty)$

  Set-Builder Notation:

  $\{x \in \mathbb{R} \mid x > a\}$

  Description:

  All real numbers strictly greater than $a$. Extends infinitely to the right.

  Number Line:

  
  (Imagine a number line with an open circle at 'a' and shading extending indefinitely to the right.)

• $[a, \infty)$

  Set-Builder Notation:

  $\{x \in \mathbb{R} \mid x \ge a\}$

  Description:

  All real numbers greater than or equal to $a$. Includes $a$ and extends infinitely to the right.

  Number Line:

  
  (Imagine a number line with a closed circle at 'a' and shading extending indefinitely to the right.)

• $(-\infty, b)$

  Set-Builder Notation:

  $\{x \in \mathbb{R} \mid x < b\}$

  Description:

  All real numbers strictly less than $b$. Extends infinitely to the left.

  Number Line:

  
  (Imagine a number line with an open circle at 'b' and shading extending indefinitely to the left.)

• $(-\infty, b]$

  Set-Builder Notation:

  $\{x \in \mathbb{R} \mid x \le b\}$

  Description:

  All real numbers less than or equal to $b$. Includes $b$ and extends infinitely to the left.

  Number Line:

  
  (Imagine a number line with a closed circle at 'b' and shading extending indefinitely to the left.)

• $(-\infty, \infty)$

  Set-Builder Notation:

  $\{x \in \mathbb{R} \mid -\infty < x < \infty\}$

  Description:

  This represents the set of all real numbers, $\mathbb{R}$. It extends infinitely in both directions.

  $(-\infty, \infty) = \mathbb{R}$

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