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Let A and B be sets of 7 elements and 10 elements, respectively. (a) How many different functions possible from A to B? from B to A? (b) How many different relations possible from A to B? (c) How many of the functions from A to B are one-to-one? (d) How many of the functions from B to A are onto?

(a) I am not sure how to approach this question. My guess is that since we could have a function per element and multiple elements and a function for all elements from A to B there could be 7+6+5+4+3+2+1 = 28. From B to A there could be 10+9+8+7+6+5+4+3+2+1.

(b) I think there can be a max of 7 relations

(c) I would also think there are 7 but at this point and beyond.

SPD
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2 Answers2

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Hint for a) for a function from A to B, how many choices of the image of the first element of A are there? Each element of A has that many choices and the choices are independent, so you multiply. for b) look up the definition of a relation. It will give you a simple expression that is a large number. For c) do like a), but each successive element has fewer choices because you can't reuse any elements in B.

Ross Millikan
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A) for $A\rightarrow B$, $10^7$ functions possible (for one element in $A$, there are $10$ possible images in $B$)
B) for $B\rightarrow A$ , $7^{10}$ functions are possible. (same logic)

C)Only $7!$ functions, one element=one image only

D)For this, follow

The Dead Legend
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    Item B is answering the second half of A. C is not correct. The first element of A that you assign a value to has $10$ choices. – Ross Millikan May 04 '17 at 03:58