Logical equivalence
Two statements are worded differently. How can you tell they say exactly the same thing?
Where you are 3 inside
What this area is about
Two statements are logically equivalent, written P \equiv Q, when their truth tables agree on every row: whatever truth values the letters take, the two are true together and false together. Equivalent statements can be swapped for each other anywhere, so an awkward statement can be traded for an easier one without changing what is claimed.
How these fit together
The test. Equivalence is checked by building both truth tables and comparing the final columns. A single row where they differ is enough to show two statements are not equivalent.
Three results. Each is one equivalence worth knowing by heart. De Morgan’s laws negate an “and” or an “or”: the connective flips and the “not” moves inside, as in \neg(P \land Q) \equiv \neg P \lor \neg Q. A conditional and its contrapositive shows p \to q says the same as \neg q \to \neg p, while the converse does not. Saying a conditional without if-then rewrites P \to Q as \neg P \lor Q, which makes a conditional’s negation routine.
A path through it
De Morgan’s laws first, then the contrapositive, then the conditional as an “or”, which leans on De Morgan to negate it.
These equivalences are what justify proof by contraposition, they return as the rule for negating a quantified statement, and De Morgan’s laws reappear almost word for word as De Morgan’s laws for sets.