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. IfXAB,thenXAorXB.
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. IfnN,then 12 14 18 116 12n 4+2n+1
(Note we can write this using a product formula: notation. Similar to sigma notation for sums.