From Statements to Predicates

Intro to Proofs · Axiom Academy

A database of twenty integers, and the leap from a single yes/no test to claims about all of them or at least one . One yes/no test, then a claim about everything Here is a database of twenty integers. A predicate P(x) is just a yes/no test you can run on any one of them — "is x prime?", "is x even?" — answering true or false for each input. The real power of mathematics shows up when you stop asking about one number and start claiming something about all of them at once, or insisting that at least one exists. You don't have to take it on faith — first watch a "for all" claim get tested, then drive the ideas yourself. Watch the sweep test the claim "every number greater than 10 is prime." It checks each number in range, left to right — 11 is prime ✓ — but the moment it reaches a single failing number, that one counterexample kills the whole claim. A statement about all objects survives only if every one passes. A "for all" claim is a promise about every object — and a single counterexample breaks the promise. A predicate is a question with a true/false answer Pick a predicate, then slide the pointer across the database. For the number under the pointer you get a verdict — P(x) = TRUE or FALSE — and every number in the whole database that satisfies P lights up. That is all a predicate is: a test that sorts the objects into the ones that pass and the ones that don't. Feed in a number, get true or false back — P(x) is a function whose answers are just "yes" and "no".

This is the written version of the interactive lesson above. See the full Intro to Proofs course.