AI Tutor
If it is raining, then the sidewalk is wet. But if you know it is not raining, can you guarantee the sidewalk is not wet?
Not quite — someone could have just turned on a garden sprinkler! To avoid logical traps like this in geometric proofs, we use two powerful variations: the inverse and the contrapositive.

Building the Inverse

A standard conditional statement has the form "If pp, then qq", written symbolically as p→qp \to q, where pp is the hypothesis and qq is the conclusion.
To form the inverse, you negate both parts without swapping their order: "If not pp, then not qq", written as ∼p→∼q\sim p \to \sim q (where the symbol ∼\sim means negation, or "not").
What happens if we both swap and negate the statements at the same time?

Building the Contrapositive

To form the contrapositive, you negate both statements AND swap their positions: "If not qq, then not pp", written as ∼q→∼p\sim q \to \sim p.
For example, the contrapositive of "If an angle measures 90∘90^\circ, then it is a right angle" is "If an angle is not a right angle, then its measure is not 90∘90^\circ".
Why do some of these related statements always seem to share the exact same truth value?

Logical Equivalence