Section 3.1

• 4. The statement beginning with "Hence" does not follow from what came before.
• 6. trivial proof
• 8.
• 10. The sum of a/b and c/d is (ad+bc)/(bd). Both numerator and denominator are integers, so result is rational.
• 16. False. Let n = 3.
• 20. There are four possible remainders. Each case is handled as this one. Suppose n = 5k + 1. n^2 = (5k+1)^2 = 25k^2 + 10k + 1 = 5(5k^2 + 2k) + 1. The remainder when this is divided by 5 is 1.
• 22. First prove directly that if n is even then 7n+4 is even. Then prove indirectly that if 7n+4 is even, then n is even.
• 24. For the "if" part: If m = n, then m^2 = n^2. If m = -n, then m^2 = (-n)^2 = n^2. For the "only if" part: suppose m^2 = n^2. From this we find (m+n)(m-n) = 0, so (m+n) or (m-n) must be zero, so m = -n or m = n.
• 28. False. The left-hand side can be larger than m, but the right-hand side cannot.
• 32. Consider all groups of three consecutive numbers. Each number gets used three times, the sum of the sums of the ten groups must be three times the sum of the numbers, or 3(55) = 165. This is an average of 165/10 = 16.5. Exercise 31 says at least one sum must be treater than or equal to 16.5, and since the sums are whole numbers, this means that at least one of the sums must be greater than or equal to 17.
• 38. To show all hummingbirds are small, let Tweety be an arbitrary hummingbird. We must show that Tweety is small. First premise implies that if Tweety is a hummingbird, then Tweety is richly colored. Therefore, by modus ponens, we can conclude that Tweety is richly colored. The third premise implies that if Tweety does not live on honey, then TWeety is not richly colored. Therefore, by modus tollens, we can conclude that Tweety does live on honey. Finally, the second premise implies if Tweey is a large bird, then Tweey does not live on honey. Therefore again by modus tollens we can now conclude that Tweety is not a large bird, i.e., that Tweey is small, as desired.
• 42. True for n <= 5.
• 48. Let d = "logic is difficult", s = "many students like logic", e = "mathematics is easy". a) s -> ~e b) fails when s is T and e is F.