Deductive Reasoning and Proof
Learn to construct logical arguments using conditional statements, contrapositives, proof structures, and counterexamples.
Logical Statements and Conditionals
A statement (or proposition) is a sentence that is either true or false, but not both. A conditional statement takes the form:
If P, then Q
Written symbolically as: P ⇒ Q
Here, P is the hypothesis (or antecedent) and Q is the conclusion (or consequent).
Related Statements
Proof Structures
A mathematical proof is a logical argument that establishes the truth of a statement beyond doubt. Common proof techniques include:
Direct Proof
Start with the hypothesis P and use logical steps (definitions, axioms, previously proven theorems) to arrive at the conclusion Q.
Proof by Contradiction
Assume the negation of what you want to prove. Show this leads to a logical contradiction, thereby proving the original statement.
Proof by Contrapositive
Instead of proving P ⇒ Q, prove the equivalent contrapositive: ¬Q ⇒ ¬P.
Disproof by Counterexample
To disprove a universal statement ("for all..."), find a single counterexample where the statement is false.
The Power of Counterexamples
A counterexample is a specific case that shows a general statement is false. You only need one counterexample to disprove a universal claim.
Claim: "All prime numbers are odd."
Counterexample: 2 is a prime number and it is even.
Conclusion: The claim is false. One counterexample is sufficient to disprove it.
Key Vocabulary
Conditional Statement
An "if...then..." statement of the form P ⇒ Q, where P is the hypothesis and Q is the conclusion.
Contrapositive
The statement ¬Q ⇒ ¬P, which is logically equivalent to the original conditional P ⇒ Q.
Counterexample
A specific case that demonstrates a universal statement is false. Only one is needed to disprove a claim.
Deductive Reasoning
Reasoning from general principles to specific conclusions using a chain of logical steps.
Worked Examples
Write the converse, inverse, and contrapositive of: "If a number is divisible by 6, then it is divisible by 3."
Original: If divisible by 6, then divisible by 3. (True)
Converse: If divisible by 3, then divisible by 6. (False -- counterexample: 9 is divisible by 3 but not 6)
Inverse: If not divisible by 6, then not divisible by 3. (False -- same counterexample: 9)
Contrapositive: If not divisible by 3, then not divisible by 6. (True -- logically equivalent to the original)
Prove by direct proof: "The sum of two even integers is even."
Step 1: Let the two even integers be a = 2m and b = 2n, where m and n are integers.
Step 2: Their sum is a + b = 2m + 2n = 2(m + n).
Step 3: Since m + n is an integer, 2(m + n) is even by definition.
Conclusion: Therefore, the sum of two even integers is always even. QED
Disprove: "For all integers n, n² + n + 41 is prime."
Strategy: Find a counterexample -- one value of n where the expression is not prime.
Try n = 40: 40² + 40 + 41 = 1600 + 40 + 41 = 1681 = 41 × 41 = 41².
Check: 1681 = 41² is not prime (it has factors 1, 41, and 1681).
Answer: The statement is false. The counterexample n = 40 gives n² + n + 41 = 41², which is composite.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the contrapositive of "If it is raining, then the ground is wet"?
Question 2
Which of the following is a valid counterexample to "All square numbers are even"?
Question 3
In a proof by contradiction, what is the first step?
Question 4
The converse of "If n is a multiple of 4, then n is even" is:
Question 5
Which statement is logically equivalent to the original conditional P ⇒ Q?
Key Concepts Summary
- ● A conditional statement has the form "If P, then Q" (P ⇒ Q).
- ● The contrapositive (¬Q ⇒ ¬P) is logically equivalent to the original; the converse and inverse are not.
- ● Direct proof uses a chain of logical steps from hypothesis to conclusion.
- ● Proof by contradiction assumes the negation and derives a contradiction.
- ● A single counterexample is sufficient to disprove a universal statement.