ass1 - n-a pairs in Ω with first element equal to A Now...

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Due: Friday, May 13 Math 239 Assignment 1 Instructions: Your assignment must be in the appropriate drop box outside the tutorial centre by noon Friday. It is your responsibility to understand what your professor views as acceptable collaboration. If your solution is the same as (or just very similar to) someone else’s, there will be problems. And if you do not put your name on the assignment, there is no point to putting it in the drop box. 1. (3 points) A function is an injection , or is one-to-one , if it takes different values on different inputs. (So x 2 viewed as a function from R to R is not injective.) Let M and N respectively be sets of size m and n . How many injections are there from M to N ? 2. (5 points) Show that the number different sequences of length 2 m we can form from the elements of {1,. .., m } such that each element occurs exactly twice is (2 m )!/2 m . 3. (7 points) Let N be a set of size n . Let Ω be the set of all ordered pairs ( A , B ) where A , B N and A B . Prove in two ways that | Ω |= 3 n : (a) Show that if | A | = a then there are 2
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Unformatted text preview: n-a pairs in Ω with first element equal to A . Now use the binomial theorem. (b) Describe a bijection between the elements of Ω and the sequences of length n with elements from {0,1,2}. 4. (6 points) Let N ≥ denote the set of nonnegative integers. (a) If the weight of the integer i is i if 3 divides i and i-1 otherwise, what is the generating function Φ N ≥ ( x )? (b) Compute the generating function if the weight of i is i , i-1 or i-2 according to whether i is congruent modulo 3 to 0, 1 or 2. 5. (7 points) If n , k ≥ 1 then ˆ n k ! = ˆ n-1 k ! + ˆ n-1 k-1 ! . Prove this in two ways: (a) The left side is the number of k-element subsets of a set of size n . Explain how we can interpret each of the terms on the right as numbers of subsets, and hence deduce the result. (b) By using the binomial theorem to expand each side of the equation (1 + x ) n = (1 + x )(1 + x ) n-1 ....
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