Question

If $A \times B = \\{ (a,x),(a,y),(b,x),(b,y)\\}$. Find A and B.

Answer 1

At first using the formula$P \times Q = \\{ (p,q):p \in P,q \in Q\\}$ we will find elements that belongs to sets A and B respectively. Then using those elements we will find the particular set i.e. A and B and finding those sets is the task we need to get the answer.

Complete step by step solution: Given data: $A \times B = \\{ (a,x),(a,y),(b,x),(b,y)\\}$
Now we know that if we have two sets let P and Q then
$(P \times Q)$ is defined as a relation, $(P \to Q)$ where the elements of $(P \times Q)$ will be in the form (p,q) where
$p \in P$ and $q \in Q$.
$P \times Q = \\{ (p,q):p \in P,q \in Q\\}$
Therefore from the given data i.e. $A \times B = \\{ (a,x),(a,y),(b,x),(b,y)\\}$
We can say that $a,b \in A$ and, $x,y \in B$
Therefore from the above statement, we can say that

$A = \\{ a,b\\}$, And $B = \\{ x,y\\}$

Note: We know that if two sets let X and Y have m and n numbers of elements respectively then the number of elements in the sets $(X \times Y)$ or $(Y \times X)$ will be the product of the number of elements in the respective sets i.e. mn.
Therefore we can also verify our answer using the above statement as
$n(A \times B) = 4$
And from the answer we have $n(A) = 2$ and, $n(B) = 2$
Since $n(B) \times n(A) = 4$ hence it satisfies the statement we mentioned.