1. When proving If p then q.”
DIRECT PROOF you need to:
CONTRAPOSITION you need to:
CONTRADICTION you need to:
2. Prove by direct proof that if m and n are integers, with modd and n is even, then 5n + m2 is odd.
3. Prove by contraposition that if x 6= 5 and is irrational,then 4x x ? 5 is irrational.
4. Prove the following existential statements by providing avalue for x. In both cases, the universe is the set of all realnumbers.
a) ?x x 2 + 5x ? 7 = 0
b) ?x x < 10 ? (x ? 2) 2 < 0
5. Prove that for any integer n, there exists an even integer kso that n < k + 1 < n + 3.
6. Prove or disprove: If x is rational and y is irrational, thenxy is irrational.
7. Prove that there is no positive integer n so that 49 < n 2< 64.
8. Prove or disprove: ?x?y ((x ? 3)y = 4x), where the universeof discourse is R for both variables.
9. Prove, by contraposition, that if the product of two realnumbers is irrational, then at least one of the two numbers isirrational. (In other words: If x · y is irrational, then x isirrational OR y is irrational.”)
10. Prove, by contradiction, that ? 3 is irrational. You may usethe Little Theorem: If m2 is a multiple of 3, then m itself is amultiple of 3
