CSIS Discrete Mathematics

5. Use mathematical induction to prove that the sum of the first n even positive integers is equal to n(n+1); in other words, that 2 +4 +6+ ... +2n = n(n + 1). 6. Consider the following true statement S: Vn e Z. if 3 divides n, then 3 divides 2n. (a) Write the negation of statement S. (b) Write the contrapositive of statement S. (c) Write the converse of statement S. (d) Write the inverse of statement S. (e) With regard to the converse of statement S: If the converse is true, then prove it, or if it is false then give a counterexample. 7. (a) Prove this statement by contradiction: av2 + b is irrational for all integers a and b where a = 0. (b) In a sentence or two, indicate why your proof from part (a) fails if we allow a = 0. 8. Let L(x,y) be the statement x loves y where the domain for x comprises all people and the domain of y comprises all desserts. Use quantifiers to express the statements below. As an example, the statement --Somebody loves ice cream is expressed by 3x L(x, ice cream). a. Everybody loves ice cream. b. Someone loves no dessert. c. No one loves prunes. d. There is some dessert that Fred does not love. e. Everyone loves some dessert. f. Everyone loves every dessert. g. Jane loves all desserts. h. Jane loves all desserts except for one. [Note (h) is more complicated than the previous ones.]