Cartesian Products of Sets

Here we will learn how to link pairs of elements from two sets and then introduce relations between the two elements in pairs.

Cartesian Products of Sets Definition

Given two non-empty sets A and B, the set of all ordered pairs $(x,y)$, where $x\in A$ and $y\in B$ is called Cartesian product of $A$ and $B$. symbolically, we write $AXB=\{(x,y):x\in A,y\in B\}$.

If $A=\{1,2,3\}$and $B=\{4,5\}$, then

$AXB=\{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}$ and

$BXA=\{(4,1),(4,2),(4,3),(5,1),(5,2),(5,3)\}$

Cartesian Product of Sets Formula

Given two non-empty sets $P$ and $Q$. The Cartesian product $PXQ$ is the set of all ordered pairs of elements from $P$ and $Q$ ,i.e.,

$PXQ=\{(p,q):p\in P,q\in Q\}$

If either $P$ or $Q$ is the null set, then $PXQ$ will also be an empty set, i.e., $PXQ=\emptyset$

What is the Cardinality of Cartesian Product?

The cardinality of Cartesian products of sets $A$ and $B$ will be the total number of ordered pairs in the $AXB$

Let $p$ be the number of elements of $A$ and $q$ be the number of elements in $B$.

So, the number of elements in the Cartesian product of $A$ and $B$ is $pq$

i.e. if $n(A)=p,n(B)=q$ then $n(AXB)=pq$

Cartesian Products of Sets Properties

(i) Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal,

i.e. $(x,y)=(u,v)$ if and only if $x=u,y=v$.

(ii) if $n(A)=p,n(B)=q$ then $n(AXB)=pq$

(iii) $A\ X\ A\ X\ A=\{(a,b,c):a,b,c\in A\}$. Here$(a,b,c)$ is called an ordered triplet.

Example:

Find the Cartesian product of three sets A = {a, b}, B = {1, 2} and C = {x, y}.

$A=\{a,b\},B=\{1,2\}$ and $C=\{x,y\}$

Solution:

The ordered pairs of$A\ X\ B\ X\ C$ can be formed as given below:

1st pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ $(a,1,x)$

2nd pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 1, y)

3rd pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 2, x)

4th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (a, 2, y)

5th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 1, x)

6th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 1, y)

7th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 2, x)

8th pair ⇒ {a, b} × {1, 2} × {x, y} ⇒ (b, 2, y)

Thus, the ordered pairs of $A\ X\ B\ X\ C$ can be written as:

$A\ X\ B\ X\ C=\{(a,1,x),(a,1,y),(a,2,x),(a,2,y),(b,1,x),(b,1,y),(b,2,x),(b,2,y)\}$