Chapter 4 Relations (Concepts)

Ordered Pairs

In mathematics, we often deal with collections of objects. When we consider a Set, the order in which the elements are listed is irrelevant. For instance, the set $\{a, b\}$ is exactly the same as the set $\{b, a\}$. However, there are numerous mathematical contexts and real-world applications where the order of elements is of paramount importance. For example, the coordinates $(2, 3)$ represent a specific point in a graph, which is distinct from the point represented by $(3, 2)$. To handle such situations where the position or order of elements matters, we use the concept of an Ordered Pair.

Definition of an Ordered Pair

An Ordered Pair is a collection of two elements, say 'a' and 'b', in which the order of the elements is specified. It is denoted by enclosing the two elements within parentheses and separating them by a comma, like $(a, b)$.

• The element 'a' is called the first component or the first element of the ordered pair.
• The element 'b' is called the second component or the second element of the ordered pair.

The term "ordered" is key here. Unlike a set, the position of each element in an ordered pair is significant.

Significance of Order vs. Sets

The fundamental difference between a set $\{a, b\}$ and an ordered pair $(a, b)$ lies solely in the importance given to the order of elements:

• For a set $\{a, b\}$, the collection is the same irrespective of the order. Thus, $\{a, b\} = \{b, a\}$.
• For an ordered pair $(a, b)$, the order matters. Thus, $(a, b)$ is generally NOT equal to $(b, a)$, unless the first and second components happen to be identical (i.e., $a=b$).

Example: The set $\{1, 5\}$ is the same as $\{5, 1\}$. But the ordered pair $(1, 5)$ is different from the ordered pair $(5, 1)$. In a coordinate system, $(1, 5)$ represents a point 1 unit along the x-axis and 5 units along the y-axis, while $(5, 1)$ represents a point 5 units along the x-axis and 1 unit along the y-axis.

Equality of Ordered Pairs

Two ordered pairs are considered equal if and only if their corresponding components are equal. This means the first element of the first ordered pair must be equal to the first element of the second ordered pair, AND the second element of the first ordered pair must be equal to the second element of the second ordered pair.

$(a, b) = (c, d) \iff a = c \text{ and } b = d$

This property is fundamental when solving problems involving unknown values within ordered pairs that are stated to be equal.

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Significance and Uses of Ordered Pairs

Ordered pairs are not merely an abstract concept; they are fundamental building blocks used across many areas of mathematics:

• Coordinate Geometry: As mentioned earlier, ordered pairs $(x, y)$ are the standard way to represent points in a two-dimensional Cartesian coordinate system. The first component gives the position along the x-axis, and the second component gives the position along the y-axis. This allows us to visually represent algebraic relationships as geometric shapes and vice versa.

• Cartesian Product: Ordered pairs are used to define the Cartesian product of two sets. The Cartesian product of sets A and B, denoted $A \times B$, is the set of all possible ordered pairs $(a, b)$ where $a$ is an element from set A and $b$ is an element from set B. This operation is a fundamental way to combine elements from different sets in an ordered manner.

• Relations: A relation between two sets A and B is formally defined as any subset of their Cartesian product $A \times B$. This means a relation is a collection of ordered pairs that link elements from A to elements from B based on some rule or condition.

• Functions: A function is a special type of relation where each element in the first set is related to exactly one element in the second set. Functions are also represented as sets of ordered pairs.

• Vectors: In vector algebra, a 2D vector can be represented by an an ordered pair $(x, y)$, indicating a displacement of $x$ units horizontally and $y$ units vertically from an origin or starting point.

Thus, the concept of ordered pairs provides the necessary structure to handle situations where the sequence or position of elements is important, paving the way for the study of relations, functions, and graphical representations.

Cartesian Product of Two Sets

Building upon the concept of ordered pairs, we can define an operation that combines elements from two sets in an ordered manner. This operation is known as the Cartesian Product, named after the French philosopher and mathematician René Descartes, whose work in coordinate geometry relies heavily on representing points using ordered pairs.

Definition of Cartesian Product

Let $A$ and $B$ be two non-empty sets. The Cartesian Product of set $A$ and set $B$ is a set that consists of all possible ordered pairs $(a, b)$ such that the first element, $a$, is taken from set $A$, and the second element, $b$, is taken from set $B$.

The Cartesian Product of $A$ and $B$ is denoted by $A \times B$, read as "A cross B".

Using Set-builder notation, the Cartesian Product is formally defined as:

$A \times B = \{(a, b) \mid a \in A \text{ and } b \in B\}$

This means that to form the set $A \times B$, you systematically pair every element from set $A$ with every element from set $B$, creating ordered pairs where the first component is always from $A$ and the second component is always from $B$.

Similarly, the Cartesian product of set $B$ and set $A$, denoted by $B \times A$, is the set of all ordered pairs $(b, a)$ where $b \in B$ and $a \in A$.

$B \times A = \{(b, a) \mid b \in B \text{ and } a \in A\}$

In general, $A \times B \neq B \times A$, because the order of elements within an ordered pair matters, and the ordered pairs in $A \times B$ have the first component from A and the second from B, while in $B \times A$ it is the other way around.

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Number of Elements in the Cartesian Product (Cardinality)

If set $A$ is a finite set with $m$ elements, and set $B$ is a finite set with $n$ elements, then the number of ordered pairs in the Cartesian product $A \times B$ is the product of the number of elements in $A$ and the number of elements in $B$.

$n(A \times B) = n(A) \times n(B)$

If $n(A) = m$ and $n(B) = n$, then $n(A \times B) = m \times n$.

Similarly, $n(B \times A) = n(B) \times n(A) = n \times m$. Since $m \times n = n \times m$, the number of elements in $A \times B$ is always equal to the number of elements in $B \times A$, even if the sets themselves are not equal.

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Cartesian Product Involving the Empty Set

If either set $A$ or set $B$ (or both) is the empty set ($\emptyset$), then their Cartesian product is also the empty set.

$A \times \emptyset = \emptyset$

$\emptyset \times B = \emptyset$

$\emptyset \times \emptyset = \emptyset$

Explanation: The definition of $A \times B$ requires forming ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. If set $A$ is empty, there are no elements $a$ to choose from. Thus, no ordered pairs can be formed, and the resulting set is empty. The same logic applies if set $B$ is empty.

Using the cardinality formula, if $n(A)=m$ and $n(\emptyset)=0$, then $n(A \times \emptyset) = n(A) \times n(\emptyset) = m \times 0 = 0$. A set with 0 elements is the empty set.

Cartesian Product of a Set with Itself

The Cartesian product of a set $A$ with itself is denoted by $A \times A$ or $A^2$. It is the set of all ordered pairs $(a_1, a_2)$ where both the first and second elements are taken from the same set $A$.

$A \times A = \{(a_1, a_2) \mid a_1 \in A \text{ and } a_2 \in A\}$

If $n(A) = m$, then the number of elements in $A \times A$ is $n(A \times A) = n(A) \times n(A) = m \times m = m^2$.

The graphical representation of $A \times A$ (when A is a set of numbers) forms points in a 2-dimensional plane, which is why coordinate systems are linked to Cartesian products.

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Cartesian Product of Three or More Sets

The concept of the Cartesian product can be extended to three or more sets. The Cartesian product of three sets $A$, $B$, and $C$, denoted by $A \times B \times C$, is the set of all possible ordered triplets $(a, b, c)$ where the first element $a \in A$, the second element $b \in B$, and the third element $c \in C$.

$A \times B \times C = \{(a, b, c) \mid a \in A, b \in B, c \in C\}$

The number of elements in the Cartesian product of three finite sets is $n(A \times B \times C) = n(A) \times n(B) \times n(C)$.

This concept generalizes to the Cartesian product of $n$ sets, which results in ordered $n$-tuples.

Relations

In mathematics, a Relation is a concept used to describe how elements from one set are connected or associated with elements from another set (or even the same set). Intuitively, a relation establishes a link or connection between pairs of elements from the sets involved. For example, "is less than", "is a factor of", "is taller than", "lives in" are all examples of relations that link elements of different sets (or the same set). Formally, relations are defined using the concept of the Cartesian product of sets.

Definition of a Relation

A Relation $R$ from a non-empty set $A$ to a non-empty set $B$ is defined as any subset of the Cartesian product $A \times B$.

Since $A \times B$ is the set of all possible ordered pairs $(a, b)$ where $a \in A$ and $b \in B$, a relation $R$ from $A$ to $B$ is simply a collection of some (or all, or none) of these ordered pairs.

Mathematically, this is written as:

$R \subseteq A \times B$

If an ordered pair $(a, b) \in A \times B$ is included in the relation $R$, it means that element $a$ from set $A$ is related to element $b$ from set $B$ according to the rule or property that defines the relation $R$. When $(a, b) \in R$, we often write this as $a R b$, which is read as "$a$ is related to $b$ by the relation $R$".

If an ordered pair $(a, b) \in A \times B$ does NOT belong to $R$ (i.e., $(a, b) \notin R$), it means that $a$ is not related to $b$ by the relation $R$, and we can write $a \not R b$.

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Domain and Range of a Relation

For a relation $R$ from set $A$ to set $B$ ($R \subseteq A \times B$):

• The Domain of the relation $R$ is the set of all the first components (elements from set $A$) of the ordered pairs that belong to $R$.

  $\text{Domain}(R) = \{a \mid (a, b) \in R \text{ for some } b \in B\}$

  The domain is always a subset of the starting set $A$ (Domain$(R) \subseteq A$).

• The Range of the relation $R$ is the set of all the second components (elements from set $B$) of the ordered pairs that belong to $R$.

  $\text{Range}(R) = \{b \mid (a, b) \in R \text{ for some } a \in A\}$

  The range is always a subset of the ending set $B$ (Range$(R) \subseteq B$).

• The entire set $B$ (the set to which the elements in the domain are related) is called the Codomain of the relation $R$. The range is always a subset of the codomain (Range$(R) \subseteq \text{Codomain}$).

Visual Representation

An arrow diagram clearly illustrates the difference between these concepts. Consider a relation R from set A to set B.

In the diagram above:

• Set A is $\{1, 2, 3, 4\}$.
• Set B is $\{a, b, c, d, e\}$.
• The Domain is the set of elements in A that are actually used as first components (i.e., have arrows leaving them). Here, Domain$(R) = \{1, 3, 4\}$. Notice element '2' is in set A but not in the domain.
• The Codomain is the entire set B, i.e., $\{a, b, c, d, e\}$.
• The Range is the set of elements in B that are actual endpoints of arrows. Here, Range$(R) = \{a, b, c\}$. Notice elements 'd' and 'e' are in the codomain but not in the range.

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Number of Possible Relations

The definition states that a relation from set $A$ to set $B$ is any subset of the Cartesian product $A \times B$.

If set $A$ has $m$ elements ($n(A) = m$) and set $B$ has $n$ elements ($n(B) = n$), then the number of elements in the Cartesian product $A \times B$ is $m \times n$.

We know that the total number of possible subsets of a set with $k$ elements is $2^k$.

Since a relation $R$ from $A$ to $B$ is any subset of $A \times B$, and $A \times B$ has $m \times n$ elements, the total number of possible relations from $A$ to $B$ is the number of subsets of $A \times B$.

$\text{Number of relations from A to B} = 2^{n(A \times B)} = 2^{n(A) \times n(B)}$

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Representing Relations

Relations can be represented in several ways, each useful in different contexts:

• Roster Form (Set of Ordered Pairs): A relation can be described by explicitly listing all the ordered pairs $(a, b)$ that satisfy the relation's property. This is the most direct way of representing a relation as a subset of the Cartesian product. This form is practical when the relation contains a small number of ordered pairs, as shown in Example 1.

  Example: If $A = \{1, 2, 3\}$ and $R = \{(1, 2), (2, 3), (3, 1)\}$ is a relation on A (from A to A).

• Set-builder Form: Similar to how sets can be defined using a property, relations can also be defined by stating the rule or condition that links the elements of the ordered pairs. This form is particularly useful for describing relations involving large or infinite sets, where listing all pairs is impossible.

  General form: $R = \{(a, b) \in A \times B \mid \text{property relating } a \text{ and } b \text{ is true}\}$

  Example: Let $A = \mathbb{Z}$ (integers). The relation $R = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} \mid y = x+1\}$ describes the "is one greater than" relation on the set of integers. Some pairs in this relation are $(0, 1), (5, 6), (-3, -2)$.

• Arrow Diagram: An arrow diagram provides a visual representation of a relation between two sets, especially when the sets are finite. We draw two distinct shapes (usually ovals or rectangles) representing the sets A and B. Elements of the sets are written inside these shapes. For every ordered pair $(a, b)$ that belongs to the relation $R$, we draw an arrow starting from element $a$ in set A and pointing to element $b$ in set B.

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Given sets $A = \{1, 2, 3\}$ and $B = \{2, 4, 6\}$. The relation $R$ is from A to B with the rule "$a$ divides $b$".

First, let's determine the ordered pairs $(a, b)$ in $A \times B$ that satisfy the condition "$a$ divides $b$". An integer $a$ divides an integer $b$ if $b = ka$ for some integer $k$. We are checking divisibility for positive integers here.

Possible ordered pairs $(a, b)$ from $A \times B$:

$(1, 2)$: Does 1 divide 2? Yes ($2 = 2 \times 1$). So, $(1, 2) \in R$.

$(1, 4)$: Does 1 divide 4? Yes ($4 = 4 \times 1$). So, $(1, 4) \in R$.

$(1, 6)$: Does 1 divide 6? Yes ($6 = 6 \times 1$). So, $(1, 6) \in R$.

$(2, 2)$: Does 2 divide 2? Yes ($2 = 1 \times 2$). So, $(2, 2) \in R$.

$(2, 4)$: Does 2 divide 4? Yes ($4 = 2 \times 2$). So, $(2, 4) \in R$.

$(2, 6)$: Does 2 divide 6? Yes ($6 = 3 \times 2$). So, $(2, 6) \in R$.

$(3, 2)$: Does 3 divide 2? No.

$(3, 4)$: Does 3 divide 4? No.

$(3, 6)$: Does 3 divide 6? Yes ($6 = 2 \times 3$). So, $(3, 6) \in R$.

The relation $R$ in roster form is $R = \{(1, 2), (1, 4), (1, 6), (2, 2), (2, 4), (2, 6), (3, 6)\}$.

Now, we draw the arrow diagram:

Draw an oval representing set A with elements 1, 2, and 3 inside it.

Draw another oval representing set B with elements 2, 4, and 6 inside it.

Draw arrows for each ordered pair in R:

• From 1 in A to 2 in B.
• From 1 in A to 4 in B.
• From 1 in A to 6 in B.
• From 2 in A to 2 in B.
• From 2 in A to 4 in B.
• From 2 in A to 6 in B.
• From 3 in A to 6 in B.

The arrow diagram visually shows which elements of A are related to which elements of B.

Domain and Range from Arrow Diagram:

Domain is the set of elements in A from which arrows originate: {1, 2, 3}. (In this case, Domain(R) = A)

Range is the set of elements in B to which arrows point: {2, 4, 6}. (In this case, Range(R) = B = Codomain)