Consider the sets A = {5n+3 | n ≤N}, B = {3n+2 | n ≤ N}. (a) List three elements in A. (b) List three elements in B. (c) Is it true that AC B? And is it true that BCA? Can you explain why? (d) How many elements do you think there are in AUB? Can you explain why you think so? (e) How many elements do you think there are in An B? Can you explain why you think so?

15. Which of the following division statements are equivalent to (6) 3 = -18? (i) 18 + 3 = 6 (ii) 18 3 = 6 (iii) 18 + 6 = -3 (iv) 18 +6 = 3 A) None B) (ii) and (iii) C) (i) and (iv) D) (ii) and (iv)

Which of the following is the contrapositive of the sentence " If n²+4n+5 is odd, then n is even"?

1d and 1e

1

Another Field We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from 13 2 the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √√3 nor are rational numbers). A number such 3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form 7 a+b√2 using the following steps: 7 7 3-5√2 21-35√2 21-35√2 21 - 35√2 21 35 == + √2 3+5√2 3+5√2 3-5√2 9 - 25(2) -41 41 41 = * Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and * is the ordinary multiplication operation. So, you need to show that: a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group.