the converse, inverse and contrapositive
Take a single 'if...then...' rule and you can flip it, negate it, or do both. The three rearrangements have names, and learning how their truth values relate keeps you from a very common reasoning blunder.
Start from the conditional 'if p, then q'. Its converse swaps hypothesis and conclusion: 'if q, then p'. Its inverse negates both parts: 'if not p, then not q'. Its contrapositive both swaps and negates: 'if not q, then not p'. The crucial fact is which of these must share the original's truth value. A conditional and its contrapositive are logically equivalent — always true together, always false together. The converse and the inverse are likewise equivalent to each other, but neither is equivalent to the original: a conditional can be true while its converse is false.
This is exactly where careful reasoning earns its keep. 'If a figure is a square, then it has four sides' is true, but its converse, 'if a figure has four sides, then it is a square', is false (a rectangle is a counterexample). Assuming the converse of a true statement is the classic logical error called affirming the consequent. The contrapositive's equivalence, by contrast, is genuinely useful: proving 'if not q, then not p' is a valid way to prove 'if p, then q', and this underlies proof by contradiction.
Conditional: 'If two angles are vertical, then they are equal' (true). Converse: 'If two angles are equal, then they are vertical' (false — two 40° angles need not be vertical). Inverse: 'If two angles are not vertical, then they are not equal' (false). Contrapositive: 'If two angles are not equal, then they are not vertical' (true, like the original).
A statement and its contrapositive always agree; the converse can disagree.
The converse of a true statement need not be true; assuming it is is the fallacy of affirming the consequent. Only the contrapositive is guaranteed equivalent to the original.