Week%201 - Supplement to Week 1 January 24, 2007 0.1...

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Unformatted text preview: Supplement to Week 1 January 24, 2007 0.1 Sequences and functions Let A be any set and let n be a natural number. Recall that A n stands for the set of ordered n-tuples of the form ( a 1 , a 2 , . . . , a n ), where each a i is an element of A . Here the order matters, so for example if n = 3 and A = { a, b, c } , then ( a, a, b ) is different from ( a, b, a ). Thus an ordered n-tuple of elements of A is just a sequence of n elements of A . This exactly the same thing as a function from the set { 1 , . . . , n } to the set A . Namely, to specify such a function, one just has to specify its value a i at each i = 1 , . . . n . If you have trouble with this, take n = 3 and A = { a, b, c } and try writing down all (or some) of the functions from { 1 , 2 , 3 } to A and then all (or some) of the sequences ( a 1 , a 2 , a 3 ) A 3 . More generally, if S and A are any sets, we can consider the set A S := F ( S, A ) of all functions from S to A . If S and A are finite, can you find a formula...
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