Cartesian Product of Sets

Introduction to Cartesian Products

In our daily life, we come across many patterns that characterise relations such as brother and sister, father and son, teacher and student. In mathematics also, we come across many relations such as number m is less than number n, line l is parallel to line m, set A is a subset of set B. In all these, we notice that a relation involves pairs of objects in certain order. In mathematics, a relation pairs objects from different sets in a specific order. To understand relations, we must first learn how to form all possible ordered pairs between two sets. This is known as the Cartesian Product.

An ordered pair is a pair of elements written in small brackets in a specific order, like $(p, q)$, where $p$ is from a set P and $q$ is from a set Q.

Definition of Cartesian Product

Given two non-empty sets A and B, the Cartesian product $A \times B$ is the set of all possible ordered pairs $(a, b)$ where $a \in A$ and $b \in B$. $$A \times B = \{(a, b) : a \in A, b \in B\}$$

• If either A or B is the null set ($\phi$), then $A \times B$ is also the null set.

Important Properties

• Equality of Ordered Pairs: Two ordered pairs are equal if and only if their corresponding first and second elements are equal. That is, $(a, b) = (x, y)$ if and only if $a=x$ and $b=y$.

• Number of Elements: If set A has p elements and set B has q elements, then their Cartesian product $A \times B$ will have pq elements. $$n(A) = p, n(B) = q \implies n(A \times B) = pq$$

• Infinite Sets: If either set A or B is an infinite set (and neither is empty), then $A \times B$ is also an infinite set.

• Ordered Triplets: The concept can be extended to three sets. $A \times A \times A = \{(a, b, c) : a, b, c \in A\}$. The element $(a, b, c)$ is called an ordered triplet.

• Non-Commutative: In general, $A \times B \ne B \times A$ because the order of elements in the pairs matters.

Example 1: Equality of Ordered Pairs

Question: If $(x+1, y-2) = (3, 1)$, find the values of x and y.

Solution: Since the ordered pairs are equal, their corresponding elements must be equal. $$x + 1 = 3 \implies x = 2$$ $$y - 2 = 1 \implies y = 3$$ So, the values are $x=2$ and $y=3$.

Example 2: Forming Cartesian Products

Question: If $G = \{7, 8\}$ and $H = \{5, 4, 2\}$, find $G \times H$ and $H \times G$.

Solution: By the definition of the Cartesian product: $$G \times H = \{(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)\}$$ $$H \times G = \{(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)\}$$ Note that $G \times H \ne H \times G$.

Example 3: Finding Sets from a Product

Question: If $A \times B = \{(p, q), (p, r), (m, q), (m, r)\}$, find A and B.

Solution: Set A is the set of all first elements in the ordered pairs. $$A = \{p, m\}$$ Set B is the set of all second elements in the ordered pairs. $$B = \{q, r\}$$