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For the following construct a proof for the statements provided. If the statement is false, give a counter example. Unless the question explicitly states a proof technique, feel free to prove by any established means.
1. Using a direct proof, prove the following statement.
Every odd integer is a difference of two squares. (Ex 42 − 32 = 7)
2. Suppose a ∈ Z. If a2 is not divisible by 4, then a is odd.
3. The number log2(3) is rational.
4. If ab is odd, then a3 +b3 is even.
5. Let A and B be sets. Show A ⊆ B ⇐⇒ B ∩ A = A.
6. IfX⊆A∪B,thenX⊆AorX⊆B.
7. Prove that 9 | (43n + 8) for every integer n ≥ 0.
n
8. Prove for every positive integer n, X k2k = (n − 1)2n+1 + 2
k=1
1 1 1 1 1 1 1
9. Ifn∈N,then 1−2 1−4 1−8 1−16 ··· 1−2n ≥4+2n+1
(Note we can write this using a product formula: notation. Similar to sigma notation for sums.)
Yn 1 1 1
1
k=1
1 − 2k ≥ 4 + 2n+1 if you are familiar with this
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