hw03 - points which are joined pairwise by line segments....

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In the Name of God Fall 2005 Discrete Mathematics Homework 3 Due: Aban 8 Problem 1. Prove or disprove: The symmetric diference oF sets S and T is the set oF elements that are in exactly in one oF S and T . Two expressions For symmetric diference are ( S - T ) ( T - S ), and ( S T ) - ( S T ). Prove these two expressions are equivalent. Problem 2. Consider all possible subsets oF the set { 1 , 2 , ··· , N } , which do not contain any neighboring elements. Prove that the sum oF the squares oF the products oF all numbers in these subsets is ( N + 1)! - 1 . ( Example : N = 3 . Then 1 2 + 2 2 + 3 2 + (1 . 3) 2 = 23 = 4! - 1 . ) Problem 3. Derive a minimum-cost realization oF the Four-variable Function that is equal to 1 iF exactly two or exactly three oF its variables are equal to 1; otherwise it is equal to 0. Problem 4. 2 n points are given in space. Altogether n 2 + 1 line segments are drawn between these points. Show that there is at least one set oF three
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Unformatted text preview: points which are joined pairwise by line segments. Problem 5. (a) De±ne a bijection between N and Z . (b) De±ne a bijection between N and N × N . Problem 6. Design an state machine to detect the sequences w satisFying the property ( n 1 ( w )-n ( w ) mod 5) > 2, where n 1 ( w ) and n ( w ) denote the number oF 1’s and 0’s in w , respectively. 1 Problem 7. For functions f : A → B and g : B → C , the composition of g and f , written g o f , is the function h : A → C where h ( a ) = g ( f ( a )). (a) Prove that if f and g are bijections, then so is gof . (b) Prove that if f : A → B is a bijection, then there is a bijection e : B → A , such that e o f = I A , where I A : A → A and I A ( a ) = a for all a ∈ A . 2...
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hw03 - points which are joined pairwise by line segments....

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