MAT148H1 Chap.2 Conditionals, Negation and Proof Strategies
Conditionals, Negation and Proof Strategies
Conditionals and when they fail
Most theorems take the form if HYPOTHESES, then RESULT, written P ⇒ Q. Such a statement is true when every instance that makes P true also makes Q true. It is false only when some instance has P true and Q false, and that instance is called a counterexample. When P is false, the conditional is counted as true, a situation called vacuous truth.
Checking a few instances can build confidence, but it never proves a universal claim; one counterexample, on the other hand, is enough to refute it. As an illustration, the claim that x² − 2y > 5 whenever x > 3 is refuted by x = 4 and y = 6, since 16 − 12 = 4. Adding the hypothesis y < 2 repairs it.
Converse, contrapositive and equivalence
The converse of P ⇒ Q is Q ⇒ P, and it is a different statement.
Every multiple of 6 is even, but 4 is even without being a multiple of 6, so the converse fails. The contrapositive ¬Q ⇒ ¬P is logically equivalent to the original: proving that an odd integer cannot be a multiple of 6 proves the same fact. When both directions hold, the statements are equivalent and we write P ⇔ Q, as when n counts as even exactly when n = 2k for some integer k.
Many definitions in the course, from injective functions to limits, are conditionals, so this vocabulary is used constantly.
Strategies chosen by the shape of the goal
A direct proof of P ⇒ Q adds P to the givens and makes Q the new goal; the final write-up opens with Suppose P and closes with Therefore Q.
A contrapositive proof assumes ¬Q and derives ¬P, which helps when ¬Q supplies a usable equation or inequality. Proof by contradiction assumes the whole claim fails and derives an impossibility. Mathematical induction proves a statement for every natural number by checking a base case and then showing that P(k) implies P(k + 1).
For induction, the base case and the inductive step are both required: showing that 4ᵏ − 1 is a multiple of 3 for every k ≥ 1 starts from 4 − 1 = 3, then rewrites 4ᵏ⁺¹ − 1 as 3·4ᵏ + (4ᵏ − 1) so the hypothesis appears inside the new expression. Without the base case the chain of implications has nothing to start from. Choosing among these is a matter of trying the most promising route in scratch work and switching if it stalls.
Throughout, every assertion must be justified by the givens or by earlier steps, and logical symbols belong in scratch work more than in the final prose.
Negation as a working tool
Contrapositive and contradiction proofs both require correct negations.
The negation of a universal statement is existential and vice versa, the negation of P ⇒ Q is P and not Q, and De Morgan’s laws turn the negation of an and into an or of negations, and the negation of an or into an and of negations. A common slip is to negate if P then Q as if P then not Q, which is still a conditional and says something different.
A reliable habit is to negate one piece at a time from the left and then read the result aloud to confirm it says the opposite. Parity arguments are the standard training ground: write each even number as 2k and each odd number as 2k + 1 with fresh letters, compute, and finish by exhibiting the integer that makes the result even or odd.
What this chapter covers
- 01
Truth conditions of an implication
- 02
Converse versus contrapositive
- 03
Direct proof from definitions
- 04
Proof by contrapositive and by contradiction
- 05
Counterexamples to universal claims
- 06
Mathematical induction
- 07
Negation rules for quantifiers and connectives
Worked example · free
Direct parity proof with odd inputs
- 1Unpack the hypotheses with fresh letters: a = 2k + 1 and b = 2j + 1 for integers k and j.
- 2Compute a² = 4k² + 4k + 1 and 3b = 6j + 3, so a² + 3b = 4k² + 4k + 6j + 4.
- 1Factor: a² + 3b = 2(2k² + 2k + 3j + 2). The bracket is an integer m, so a² + 3b = 2m is even.
- 1Sanity check after the proof: a = 3, b = 5 give 9 + 15 = 24, and k = 1, j = 2 give m = 12, so 2m = 24.
Key terms
- Contrapositive
- The statement ¬Q ⇒ ¬P formed from P ⇒ Q by negating and swapping both parts. It is logically equivalent to the original, so proving it proves the original claim.
- Converse
- The statement Q ⇒ P obtained by swapping the hypothesis and conclusion of P ⇒ Q. It is a separate claim that may be false even when the original is true.
- Counterexample
- A single instance in which every hypothesis of a claimed theorem holds but the conclusion fails. Finding one shows the universal claim is false, while no number of agreeing instances can show it is true.
- Induction Principle
- The rule that a statement P(n) holds for all natural numbers from a starting value once the base case is true and P(k) implies P(k + 1) for every k.
Conditionals, Negation and Proof Strategies FAQ
Is the contrapositive the same as the converse?
No. For if P then Q, the converse is if Q then P and may be false, while the contrapositive is if not Q then not P and is always equivalent to the original. Only the contrapositive can replace the original in a proof.
When should I use proof by contradiction instead of a direct proof?
Contradiction suits claims that something cannot exist or that a set is empty, where assuming an object exists gives concrete equations to work with. If assuming the hypothesis already leads straight to the goal, a direct proof is shorter and clearer.
How do I negate an if then statement?
The negation of if P then Q is P and not Q: the hypothesis holds while the conclusion fails. It is not another conditional. For example, the negation of if n is odd then n squared is odd says some odd n has an even square.
Exam move
For each proof you study, first write the givens and the goal in two columns, then decide which strategy the goal suggests before reading the solution. Rewrite one parity or divisibility proof per day from memory using fresh letters. Once a week, take a true theorem, state its converse, and either prove it or find a counterexample. Keep a running list of flawed proofs you have seen, with one line on where each went wrong.
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