1. Prove: log 2 7 is irrational 2. Prove: If x is a real number, then | x + 1 | ≤ 3 implies that − 4 ≤ x ≤ 2. 3. Consider the following theorem: “If m2 is odd, then m is odd.” Indicate what, if...

1. Prove: log₂7 is irrational

2. Prove: If x is a real number, then | x + 1 | ≤ 3 implies that − 4 ≤ x ≤ 2.

3. Consider the following theorem: “If m2 is odd, then m is odd.” Indicate what, if anything, is wrong with each of the following “proofs.”

(a) Suppose m is odd. Then m = 2k + 1 for some integer k. Thus m²
=(2k + 1)²
= 4k²
+ 4k + 1 = 2(2k²
+ 2k) +1, which is odd. Thus if m2 is odd, then m is odd.

(b) Suppose m is not odd. Then m is even and m = 2k for some integer k. Thus m²
= (2k)²
= 4k²
= 2(2k²), which is even. Thus if m is not odd, then m²
is not odd. It follows that if m²
is odd, then m is odd.