Converse, inverse and contrapositive

Starting from a single implication $p → q$, three related implications can be built by rearranging or negating its two parts:

• $q → p$ is the converse of $p → q$ (the sides are swapped).
• $~p → ~q$ is the inverse of $p → q$ (both sides negated, order kept).
• $~q → ~p$ is the contrapositive of $p → q$ (both sides negated AND swapped).

Activity. Building the truth table for all four side by side settles which pairs are secretly the same statement:

The p→q column (T,F,T,T) matches the ~q→~p column exactly, so $p → q ≡ ~q → ~p$ — an implication and its own contrapositive always carry the same truth value. Separately, the q→p column (T,T,F,T) matches the ~p→~q column exactly, so $q → p ≡ ~p → ~q$ — the converse and the inverse are always equivalent to each other. But this second pair is a genuinely different statement from the original implication, since its column (T,T,F,T) does not match p→q's column (T,F,T,T). This is exactly why proving a contrapositive is an accepted way to establish the original implication, while proving a converse or an inverse is not — that proves a related but logically different claim.

Ex.1 — write the negations.

i) '3+3<5 or 5+5=9' is $p ∨ q$; its negation is $~p ∧ ~q$, i.e. '3+3 ≥ 5 and 5+5 ≠ 9'.

ii) '7>3 and 4>11' is $p ∧ q$; its negation is $~p ∨ ~q$, i.e. '7 ≤ 3 or 4 ≤ 11'.

iii) 'The number is neither odd nor a perfect square' means $~p ∧ ~q$ (not odd, and not a perfect square); its negation is $~(~p ∧ ~q) ≡ p ∨ q$, i.e. 'the number is odd or a perfect square'.

iv) 'The number is even if and only if it is divisible by 2' is $p ↔ q$; using $~(p ↔ q) ≡ (p ∧ ~q) ∨ (q ∧ ~p)$, the negation is 'the number is even but not divisible by 2, or it is divisible by 2 but not even'.

Ex.2 — negate the quantified statements. Negating a quantified statement swaps the quantifier — 'all'/'for every' becomes 'some'/'there exists' and vice versa — while the condition itself also gets negated.

i) 'All natural numbers are rational' negates to 'some natural numbers are not rational'.

ii) 'Some students of class X are sixteen years old' negates to 'no student of class X is sixteen years old'.

iii) '$∃n ∈ N$ such that $n+8>11$' negates to '$∀n ∈ N,\ n+8 ≤ 11$'.

iv) '$∀x ∈ N,\ 2x+1$ is odd' negates to '$∃x ∈ N$ such that $2x+1$ is not odd'.

Ex.3 — converse, inverse and contrapositive.

(1) Let p: 'a function is differentiable', q: 'a function is continuous', so the given statement is $p → q$.

• Converse ($q→p$): 'If a function is continuous then it is differentiable.' …