Question

Prove by contradiction: Let a and b be integers. Show that if is odd, then a is odd and b is odd.

a) State the negation of the above implication.

b) Disprove the negation and complete your proof.

Answer #1

Statement: "For all integers n, if n2 is odd then n is odd"
(1) prove the statement using Proof by Contradiction
(2) prove the statement using Proof by Contraposition

Prove by contradiction that:
For all integers a and b, if a is even and b is odd, then 4 does
not divide (a^2+ 2b^2).

Let a, b be an element of the set of integers. Proof by
contradiction: If 4 divides (a^2 - 3b^2), then a or b is even

Prove: Let a and b be integers. Prove that integers a and b are
both even or odd if and only if 2/(a-b)

For all integers x, if (x ^2 + y^2) is not equal to 0 (mod 4),
then x is odd or y is odd. Write the contrapositive of this
statement. Write the contrapositive of this statement. Write the
negation of this statement. c. Prove this statement using a proof
by contraposition or a proof by contradiction?

Prove by contradiction that 17n + 2 is odd --> n is odd.

8. Let a, b be integers. (a) Prove or disprove: a|b ⇒ a ≤ b. (b)
Find a condition on a and/or b such that a|b ⇒ a ≤ b. Prove your
assertion! (c) Prove that if a, b are not both zero, and c is a
common divisor of a, b, then c ≤ gcd(a, b).

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

If a and b are odd integers, then 3a + 2b is an odd integer.
Construct a know show table and formal proof.

7. Prove by contradiction or contrapositive that for all
integers m and n, if m + n is even then m and n are both even or m
and n are both odd.

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