Question

If A,B are two sets such that n(A×B)=6 and some element of A×B are (-1,2),(2,3),(4,3),then find A×B

Answer 1

We have,

$\textsf{If}\\ \\ n(A)=a\\ \\ \textsf{and}\\ \\ n(B)=b,\\ \\ \textsf{then}\\ \\ n(A\times B)=ab$

Here, given that  $n(A\times B)=6$

6 can be factorized as,

$6=1\times 6\\ \\ 6=2\times 3$

So, either of the following is true that:

$1.\ \ n(A)=1\ \ \ ; \ \ \ n(B)=6\\ \\ 2.\ \ n(A)=2\ \ \ ; \ \ \ n(B)=3\\ \\ 3.\ \ n(A)=3\ \ \ ; \ \ \ n(B)=2\\ \\ 4.\ \ n(A)=6\ \ \ ; \ \ \ n(B)=1$

But some elements of  $A\times B$  are given that,

$\{(-1,\ 2),\ (2,\ 3),\ (4,\ 3)\}\ \subset\ (A\times B)$

From this, we can find out that,

$\{-1,\ 2,\ 4\}\subset A\\ \\ \{2,\ 3\}\subset B$

Each subsets have a cardinality of 3 and 2 respectively, so we can say that,

$A=\{-1,\ 2,\ 4\}\\ \\ B=\{2,\ 3\}$

Hence,

$\large \text{ \bold{A\times B=\{(-1,\ 2),\ (-1,\ 3),\ (2,\ 2),\ (2,\ 3),\ (4,\ 2),\ (4,\ 3)\}} }$

Hence found!