Converse, Inverse, Contrapositive — The Four-Form Carousel

"If a number is divisible by 6, then it is divisible by 3." That is an implication. Now take it apart and rebuild it three different ways:

Swap the hypothesis and conclusion → you get the converse.
Negate both parts, keeping the order → you get the inverse.
Swap and negate → you get the contrapositive.

Four forms. Four truth tables. Which of them match the original? The carousel below walks through all four and shows exactly which forms are logically equivalent — and which ones are impostors in disguise.

The four forms at a glance

Converse, inverse, contrapositive

Starting from the implication p \Rightarrow q:

Converse: q \Rightarrow p — swap hypothesis and conclusion.
Inverse: \lnot p \Rightarrow \lnot q — negate both, keep the order.
Contrapositive: \lnot q \Rightarrow \lnot p — swap and negate.

Two of these four are logically equivalent to the original and two are not. The carousel shows which.

Drag through the four forms

The four forms of "if $p$ then $q$." Drag the slider through each card. Green cards (original and contrapositive) share the truth-value pattern $T, F, T, T$. Red cards (converse and inverse) share the opposite pattern $T, T, F, T$. That shared-pattern observation is the entire theorem: **original $\equiv$ contrapositive**, and **converse $\equiv$ inverse**, but crossing the pairs gives two completely different statements.

Work it on a concrete sentence

Take the true statement: "If a natural number n is divisible by 6, then n is divisible by 3." Set p = "n is divisible by 6" and q = "n is divisible by 3."

Original (p \Rightarrow q): If n is divisible by 6, then n is divisible by 3. True. (Every multiple of 6 is also a multiple of 3.)
Converse (q \Rightarrow p): If n is divisible by 3, then n is divisible by 6. False. (n = 9 is divisible by 3 but not by 6 — a counterexample.)
Inverse (\lnot p \Rightarrow \lnot q): If n is not divisible by 6, then n is not divisible by 3. False. (Again n = 9: not divisible by 6, still divisible by 3.)
Contrapositive (\lnot q \Rightarrow \lnot p): If n is not divisible by 3, then n is not divisible by 6. True. (If a number is not a multiple of 3, it cannot be a multiple of 6.)

The original and the contrapositive are both true. The converse and the inverse are both false. That is the pattern the truth table predicts.

The pairing rule

p \Rightarrow q \ \equiv\ \lnot q \Rightarrow \lnot p \qquad \text{(original and contrapositive)}

q \Rightarrow p \ \equiv\ \lnot p \Rightarrow \lnot q \qquad \text{(converse and inverse)}

Why the pairs match: the contrapositive is obtained by applying "swap and negate," and so is "swap the inverse." In truth-table terms, both operations flip the single F entry in the original's T, F, T, T pattern to a different position. Carrying out both transformations lands you back where you started. The shortcut to remember: contrapositive = original, converse = inverse, but the two groups are not equivalent to each other.

A one-line test to stop confusing them

Given an implication, label the two parts. "Swap" moves you to the converse. "Negate" moves you to the inverse. "Swap and negate" moves you to the contrapositive. If a proof starts "Suppose q is false…" then you are doing the contrapositive — which is allowed because contrapositive is equivalent to the original. If a proof starts "Suppose q is true…" then you are proving the converse, which is a different theorem. Many failed JEE answers come from students who silently proved the converse and claimed the original.

Where this matters

Proof by Contrapositive relies entirely on the fact that the contrapositive is logically equivalent to the original — so a proof of \lnot q \Rightarrow \lnot p is a legitimate proof of p \Rightarrow q. The converse, despite looking close, is a separate theorem that needs its own proof if you want it.