If–then

A parent promises: “If you finish your homework, then we’ll go for ice cream.” When has the parent broken the promise?

• Homework done, ice cream: promise kept.
• Homework done, no ice cream: promise broken.
• Homework not done, no ice cream: promise kept. Nothing was promised for this case.
• Homework not done, ice cream anyway: still kept! The promise only said what happens if the homework gets done. A generous parent didn’t break anything.

That’s the whole meaning of “if $p$ then $q$”: the only way to break it is $p$ happening without $q$. Mathematicians use exactly this meaning, and once it clicks, a lot of proofs become much easier to read.

The real words

• Implication ($p \to q$): $p$ is the hypothesis, $q$ is the conclusion. It’s false only when $p$ is true and $q$ is false.
  • When $p$ is false, $p \to q$ is vacuously true. “If pigs fly, I’m the queen” is technically true.
  • Many ways to say it: ”$p$ implies $q$”, ”$q$ if $p$”, ”$p$ only if $q$”, ”$p$ is sufficient for $q$”, ”$q$ is necessary for $p$”.
• Converse and contrapositive of $p \to q$:
  • Converse: $q \to p$. Not the same! “If it’s a dog, it’s a mammal” is true; “if it’s a mammal, it’s a dog” isn’t.
  • Contrapositive: $\lnot q \to \lnot p$. Always the same as $p \to q$. “If it’s not a mammal, it’s not a dog.” Lesson 9 turns this into a proof technique.
  • Inverse: $\lnot p \to \lnot q$. Same as the converse, so also not the original claim.
• If and only if ($\leftrightarrow$): $p \leftrightarrow q$ is true when $p$ and $q$ match. It means $p \to q$ and $q \to p$. Mathematicians write “iff”. Proving an iff takes two proofs, one for each direction.
• In code, $p \to q$ is (not p) or q.