Chapter 6: The Contrapositive

Question 1

What is the logical equivalence of an implication?

Answer 1

Question 2

Examples of contrapositive statements?

Answer 2

Question 3

What should match about an implication and its contrapositive?

Answer 3

• Also the contrapositive of the contrapositive is logically equivalent to the implication
• ~Q –> ~P is ~~P –> ~~Q which is P–> Q

Question 4

What is the general structure of a proof using the contrapositive?

Answer 4

Question 5

PROOF: Suppose n ∈ N. IF n^2 is odd, then n is odd.

Answer 5

Question 6

PROOF SKETCH: Suppose n ∈ N. Then, n is odd if and only if 3n+5 is even.

Answer 6

Question 7

PROOF: Suppose n ∈ N. Then, n is odd if and only if 3n+5 is even.

Answer 7

Question 8

How can PART 2 of the following proof be treated as a direct proof instead of a contrapositive?

PROOF: Suppose n ∈ N. Then, n is odd if and only if 3n+5 is even.

Answer 8

Question 9

PROOF: Let a,b ∈ Z< and let p be a prime. If p ∤ ab, then p ∤ a and p ∤ b.

Answer 9

Question 10

When can you use the “without loss of generality, assume x” in a proof?

PROOF: Let a,b ∈ Z< and let p be a prime. If p ∤ ab, then p ∤ a and p ∤ b.

Answer 10

When you have multiple cases that are identical mathematically but just swap out some variable names

Question 11

Suppose a, b, n ∈ N. If 36a 6≡/ 36b (mod n), then n ∤ 36.

Answer 11

The fact that this proposition says a lot of things are not happening is
one indication that the contrapositive could be worthwhile

Question 12

What are two lemma and the squares of any integer?

Answer 12

Question 13

PROOF: If a is an odd integer, then x²+ x − a² = 0 has no integer solution

Answer 13