Proof Techniques: Direct, Contrapositive & Contradiction (Discrete Mathematics)
Free printable Discrete Mathematics worksheet on choosing and applying proof techniques: direct proof, proof by contrapositive, and proof by contradiction, plus writing the contrapositive, converse, and negation of a conditional statement.
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Discrete Mathematics · Math worksheet
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Proof Techniques: Direct, Contrapositive & Contradiction
Choose and apply the right proof technique. A direct proof assumes the hypothesis and derives the conclusion. A contrapositive proof shows "if not Q then not P" instead. A contradiction proof assumes the statement is false and derives an impossibility. Remember: the contrapositive is equivalent to the original statement; the converse is not.
- 1.Which proof technique is the standard first choice for proving: "If n is an odd integer, then n² is odd."
- a) Direct proof
- b) Proof by contrapositive
- c) Proof by contradiction
- d) Proof by induction
- 2.Which proof technique is the standard first choice for proving: "If 3n + 2 is odd, then n is odd."
- a) Direct proof
- b) Proof by contrapositive
- c) Proof by contradiction
- d) Proof by induction
- 3.Which proof technique is the standard first choice for proving: "There are infinitely many prime numbers."
- a) Direct proof
- b) Proof by contrapositive
- c) Proof by contradiction
- d) Proof by induction
- 4.Which proof technique is the standard first choice for proving: "For every integer n ≥ 1, ≥ n + 1."
- a) Direct proof
- b) Proof by contrapositive
- c) Proof by contradiction
- d) Proof by induction
- 5.State the contrapositive of: "If n is even, then n² is even."
- 6.State the converse of: "If x > 5, then x > 2."
- 7.State the negation of: "If a graph is a tree, then it has no cycle."
- 8.In the direct proof that the sum of two odd integers is even, odd integers are written m = 2a + 1 and n = 2b + 1, so m + n = 2(a + b + 1). Check the algebra with a = 5 and b = 4: compute m, n, and m + n.
- 9.In the direct proof that the sum of two odd integers is even, odd integers are written m = 2a + 1 and n = 2b + 1, so m + n = 2(a + b + 1). Check the algebra with a = 8 and b = 9: compute m, n, and m + n.
- 10.The direct proof that the product of two odd integers is odd expands (2a + 1)(2b + 1) = 2(2ab + a + b) + 1. Check it with a = 4 and b = 8: compute the product (2a + 1)(2b + 1).
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