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hw4solutions - EE 369 Homework 4 Solutions(Sixth Edition...

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EE 369 Homework 4 Solutions (Sixth Edition) Section 2.1 2. Given P-->Q then Q'-->P' is the contrapositive and Q-->P is the converse. The inverse is P'-->Q'. To which of the other three is the inverse equivalent? P'-->Q' is the contrapositive of the converse Q-->P, so the inverse and the converse are equivalent. 3. Provide counterexamples: a. Every geometric figure with 4 right angles is a square. counterexample: A nonsquare rectangle. b. If a real number is not positive,then it must be negative. counterexample: 0 c. All people with red hair have green eyes or are tall. counterexample: a short,blue eyed person with red hair. d. All people with red hair have green eyes and are tall. counterexample: a short person with red hair 13. Prove that the sum of even integers is even..(do a proof by contradiction) Given n even integers x_1,x_2,...,x_n, assume that x_1+x_2+...+x_n=S is odd, then we know from the definition of odd and even numbers that x_1=2*k_1, x_2=2*k_2,...x_n=2*k_n for some integers k_1,k_2,...k_n, and S=2*k+1 for some integer k. We then have x_1+x_2+...+x_n=2(k_1+k_2+...+k_n)=S=2*k+1 and we can get 2(k_1+k_2+...+k_n-k)=1, where k_1+k_2+...+k_n-k is some integer. This is a contradiction since 1 is not even. 21. If a number x is positive, so is x+1. (do a proof by contraposition) The contrapositive is: if x+1<=0 then x<=0. If x+1<=0, then x<=-1<0,so x<0 and therefore x<=0. 26. If x is an even prime number then x = 2. Let x be a prime number with x=2k where k is an integer.Then both 2 and k divide x. Because x is prime, x is divisible only by itself and 1. Assume that x is some even number not equal to 2. In this case k won't be equal to 1 since that will make x equals to 2. We also know that x is divisible by k and x is not equal to k, therefore x is divisible by some integer that is not itself nor 1, and we can say that x is not a prime, and there is a contradiction. Therefore x=2.
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