Given:
The two sets are:
[tex]A=\{1,4\}[/tex]
[tex]B=\{2,3,5\}[/tex]
To find:
The [tex]A\times B[/tex] and the number of relations from A to B.
Solution:
If A and B are two sets, then
[tex]A\times B=\{(x,y)|x\in A, y\in B\}[/tex]
We have,
[tex]A=\{1,4\}[/tex]
[tex]B=\{2,3,5\}[/tex]
Then,
[tex]A\times B=\{(1,2),(1,3),(1,5),(4,2),(4,3),(4,5)\}[/tex]
If number of elements in set A is m and the number of element in set B is n, then the number of relations from A to B is [tex]2^{m\times n}[/tex].
From the given sets, it is clear that,
The number of elements in set A = 2
The number of elements in set B = 3
Now, the number of relations from A to B is:
[tex]2^{m\times n}=2^{2\times 3}[/tex]
[tex]2^{m\times n}=2^{6}[/tex]
[tex]2^{m\times n}=64[/tex]
Therefore, the required relation is [tex]A\times B=\{(1,2),(1,3),(1,5),(4,2),(4,3),(4,5)\}[/tex] and the number of relations from A to B is 64.
