AI Tutor
Imagine your parents promise: "If you clean your room, then you get ice cream." If you get ice cream, does that automatically mean you cleaned your room?
Not necessarily β€” maybe it was just your birthday! In geometry, swapping or negating parts of a rule creates completely new statements that might be true or completely false.
Every conditional statement is built from a hypothesis (pp) and a conclusion (qq). But what happens when we rearrange or negate them?

The Three Related Statements

Starting from a standard conditional statement (pβ†’qp \to q), you can create three new statements by swapping places, adding negations (the word "not", written as ∼\sim), or doing both.
The converse swaps the hypothesis and conclusion (qβ†’pq \to p), while the inverse negates both parts without swapping (∼pβ†’βˆΌq\sim p \to \sim q).
The contrapositive does both: it flips the order AND negates both parts (∼qβ†’βˆΌp\sim q \to \sim p).
Back in the 3rd century BCE, the Greek philosopher Chrysippus established propositional logic by proving that if a conditional statement is true, its contrapositive is guaranteed to be true as well.
Let's see how this works with an actual geometric shape.

Worked Example: The Square Test

Let's test the true conditional statement: "If a shape is a square (pp), then it has 4 sides (qq)."