Question

In: Advanced Math

Let A be a set with m elements and B a set of n elements, where...

Let A be a set with m elements and B a set of n elements, where m; n are positive integers. Find the number of one-to-one functions from A to B.

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Expert Solution

Let A be a set with m elements and B a set of n elements, where m,n are positive integers .

Let .

If is a one-to-one function then are distinct element of B so the set B contains at least m elements .

Let's find number of one-to-one functions from A to B .

Let be any one-to-one function .

Then have n choises as it can map to any element of B .

Also have n-1 choises as cannot map to the element of B which already mapped by .

have n-2 choises as cannot map to the element of B which already mapped by and .

and so on .

and al last have n-m+1 choises as cannot map to the element of B which already mapped by , ,..., .

Hence total number of one-to-one function from A and B is ,

Note that is well defined as

Answer : .

.If you have any doubt or need more clarification at any step pleae comment.

Colby Messinger answered 1 year ago

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