You want to prove 'If a number is divisible by 4, then it is divisible by 2' but find a direct proof difficult. Which of the following is logically equivalent and might offer a cleaner approach?
AIf a number is divisible by 2, then it is divisible by 4 — the converse
BA number is not divisible by 4 and not divisible by 2 — negation of both parts
CIf a number is not divisible by 2, then it is not divisible by 4 — the contrapositive
DA number is divisible by 4 or not divisible by 2 — a disjunction
The contrapositive of 'if p then q' is 'if not q then not p,' and they are logically equivalent — they have identical truth tables. This means proving the contrapositive is not a workaround or an approximation; it proves the original statement exactly. The converse (option A) is a different claim entirely and is not equivalent to the original. The contrapositive is a valid proof strategy specifically because of this logical equivalence.
Question 2 Multiple Choice
A student negates the statement 'It is raining and it is cold' by writing 'It is not raining and it is not cold.' Is this correct?
AYes — negation distributes directly over 'and,' so each part is negated
BNo — by De Morgan's law, the correct negation is 'It is not raining or it is not cold'
CYes — negating each part separately always gives the correct negation
DNo — negating a conjunction always requires negating the entire statement without changing the connective
De Morgan's law states that ¬(p ∧ q) ≡ ¬p ∨ ¬q. When you negate a conjunction, the 'and' becomes 'or' — the connective flips. The statement 'It is raining and it is cold' is false as soon as either condition fails, which is exactly what 'It is not raining OR it is not cold' captures. The student's error — keeping 'and' while negating each part — produces a stronger statement that requires both conditions to fail simultaneously, which is incorrect.
Question 3 True / False
Two statements are logically equivalent if there is at least one row in their truth tables where they have the same truth value.
TTrue
FFalse
Answer: False
Logical equivalence requires that the statements match in every row of their truth tables — not just some. Two completely unrelated statements might happen to be both true in some cases without being equivalent. Equivalence is a much stronger condition: P ≡ Q means P and Q have identical truth values for every possible combination of truth values of their component variables. Anything less than complete agreement across all rows is not equivalence.
Question 4 True / False
Proving the contrapositive of a conditional statement is a legitimate proof strategy because the contrapositive is logically equivalent to the original statement.
TTrue
FFalse
Answer: True
Logical equivalence means the two statements have identical truth values in every case, so any proof of one is automatically a proof of the other. The contrapositive of p → q is ¬q → ¬p, and their truth tables are identical — both are false only when p is true and q is false. This is not a trick or a shortcut but a genuine substitution: when two statements are equivalent, you can replace one with the other anywhere in a proof.
Question 5 Short Answer
Why does the equivalence p → q ≡ ¬p ∨ q explain why assuming the hypothesis in a direct proof is a valid strategy?
Think about your answer, then reveal below.
Model answer: The equivalence says 'if p then q' means the same thing as 'either p is false or q is true.' To prove this disjunction, the only interesting case is when p is true — because if p is false, the disjunction ¬p ∨ q is immediately true regardless of q. So you only need to handle the case where p holds. Assuming the hypothesis p in a direct proof is exactly this: you are focusing on the one case that requires proof, and showing q follows. The equivalence explains why this suffices.
This rewriting also explains why a conditional with a false hypothesis is vacuously true: if p is false, ¬p is true, so ¬p ∨ q is true regardless of q. The logical equivalence between implication and disjunction is the foundation for understanding why conditional proofs work the way they do.