KSET 2015 computer science

Question

Let X and Y be two sets having n elements
each. Then the total number of bijective
functions from X and Y is
(A) n (B) 2n
(C) n! (D) 2n

Comments

one one and onto function is called  as bijective

and here x and y have same cardinality

let it  is suppose  n

then first element of set x have total of ways  of  matching with set y= n

and second ……………………………………………………...y=n-1

ans so on…

then tatal=n!

Answer 1

Bijection means both Onto and One-One.

Now, if we have two sets with $X$ and $Y$ with $N$ elements each, each element from $Y$ has to map exactly $1$ element of $X$.

So, this simplifies your problem to place $N$ different balls to $N$ different boxes, where each box can hold only $1$ ball. In other way, you just have to find out how many ways you can arrange the $N$ elements of $Y$.

This is simple permutation i.e. $N!$.

Do Note – Bijection is only possible between two sets if both have same cardinality.