Question

What is the formula for no. of functions from set A to set Bif there are m elementsin A and n elementsin B? (No Spam)​

Answer 1

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In this case, choosing such a function is the same as choosing the p elements of B which are in the image of the map. Thus, the number of such maps is the number of ways to choose p elements out of q where order does not matter, or q!(q−p)!maps.

For onto maps A→B, we now need A to be at least as big as B, so p≥q.

After similar counting, we can say that the number of such maps is equal to the number of ways of breaking a p element set into q nonempty subsets, corresponding to the fibers over the elements of B.

Answer 2

Step-by-step explanation:

1. Let X and Y are two sets having m and n elements respectively.
2. In a function from X to Y, every element of X must be mapped to an element of Y.
3. Therefore, each element of X has ‘n’ elements to be chosen from.
4. Therefore, total number of functions will be n×n×n.. m times = n^m

For example:-. Let X, Y, Z be sets of sizes x, y and z respectively. Let W = X x Y. Let E be the set of all subsets of W. The number of functions from Z to E is:

sol)

As W = X x Y is given, number of elements in W is xy. As E is the set of all subsets of W, number of elements in E is 2xy. The number of functions from Z (set of z elements) to E (set of 2xy elements) is 2xyz.