課程名稱︰離散數學 課程性質︰資訊系選修 課程教師︰呂育道 開課學院：電資學院 開課系所︰資訊系 考試日期（年月日）︰101/3/29 考試時限（分鐘）：180分鐘 是否需發放獎勵金： （如未明確表示，則不予發放） 是 試題 : 12 13 Problem 1 (10 points) What is the coefficient of x y in the expansion 25 of (2x + 3y) ? Problem 2 (10 points) Show that for all positive integers m and n, m+n m+n n ( ) = (m + 1)( ) m n+1 Problem 3 (10 points) Prove that if n is a nonnegative integer, then 2n n n 2 ( ) = Σ ( ) n k=0 k Problem 4 (20 points) Let A → B denote the set of functions from set A to n set B. (a) [10 points] How many functions in {0,1,2} → {0,1,2} are there? n m (b) [10 points] How many functions in ({0,1,2} → {0,1,2}) → {0,1,2} are b b a (a ) there? (Do not write somthing like x as it is ambiguous. Write x b a or (x ) .) Problem 5 (10 points) Prove the mutinominal theorem: If n is a positive integer, then n n1 n2 nm (x1+x2+…+xm) = Σ C(n;n1,n2,…,nm)x1 x2 …xm n1+n2+…+nm=n where n! C(n;n1,n2,…,nm) = ─────── n1!n2!…nm! + Problem 6 (10 points) Prove that for any positive integer for n 屬於 Z n Fi-1 Fn+2 Σ─── = 1 - ─── i=1 i n 2 2 Problem 7 (20 points) Consider (p V q) → r 1) Give an equivalent statement without → 2) Is it a tautology? 3) Is it a contradiction? 4) Negate the result in (1) first and apply DeMorgan's laws to move the negation connective to the primitive statements p,q,r. Problem 8 (10 points) A function f : A → B is called one-to-one if each element of B appears at most once as the image of an element in A. Assume A = {a1,a2,…am}. How many one-to-one functions from A to B are there if |A| = m and |B| = n ≧ m? --

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