An implication is a promise: if the first statement holds, the second one does too. A promise is broken only when its condition is met and its conclusion fails. When the condition never arises, the promise was never tested, and mathematics counts an untested promise as kept.

A shop promises: if you spend twenty pounds, you get a free cup. Only one thing breaks that promise, namely spending twenty and being refused the cup. Spend nothing and the shop has broken nothing, whether or not it hands you a cup anyway.

We write P for the hypothesis and Q for the conclusion. Two rearrangements are easy to confuse: the converse swaps them, while the contrapositive swaps them and negates both. Only one of the two says the same thing as the original.

• The implication is $P \implies Q$.
• The converse is $Q \implies P$, and it may be false when the original is true.
• The contrapositive is $\neg Q \implies \neg P$, and it always has the same truth value as the original.

Rather than compare when the two are true, compare when they are false. Each is false in exactly one situation, and the two situations turn out to be the same one described twice.

This is the definition: the only broken promise has its condition met and its conclusion failing.

Apply the same definition to the contrapositive, whose hypothesis is $\neg Q$.

Read those two conditions back in terms of $P$ and $Q$ themselves.

The two descriptions have collapsed into one and the same case.

Each has exactly one truth value, so agreeing on falsity forces agreement everywhere.

If $2 > 1$ then $2 > 5$.

The hypothesis holds and the conclusion fails, so this implication is false. It is the only shape that fails.

If $3 \in \varnothing$ then $3 > 100$.

Nothing belongs to the empty set, so the hypothesis never holds and the implication is true.

If $n = 2$ then $n$ is even.

True. Its converse says every even number equals two, which $4$ already contradicts.

If $n$ is even then $n^2$ is even.

Its contrapositive: if $n^2$ is odd then $n$ is odd. Same claim, stated from the other end.

A necessary condition must hold for the claim to hold. A sufficient condition is enough on its own.

• $P\implies Q$ is false only when $P$ is true and $Q$ is false.
• A false hypothesis makes the implication vacuously true.
• The contrapositive always agrees; the converse need not.
• "Only if" gives the forward implication, "if" the backward one, and a biconditional needs both.

Great work! You now know how to:

• Say when an implication is false, and when it is vacuously true
• Translate "if", "only if", necessary and sufficient into arrows
• Keep the converse and the contrapositive apart
• Prove a biconditional as two implications