Proving Cancellation Laws

Intro to Proofs · Axiom Academy

EXAMPLE Proving the Cancellation Law Construct a formal proof from the group axioms alone — no assuming that "cancel the a " works the way it does in arithmetic. The group axioms (our only tools) Let G be a group and let . Prove the left cancellation law : if ab = ac , then b = c . You may use only the three group axioms above — not any property borrowed from ordinary number arithmetic. The inverse a^ -1 is the eraser: associativity slides it next to its partner, the inverse axiom turns a^ -1 a into e , and the identity axiom drops the e . Run the identical chain on the right-hand side and only c remains. You proved a "familiar" fact from scratch. Here is what made the argument work — and exactly where it would break. We proved cancellation; we didn't assume it. Nothing borrowed from number arithmetic — every move was associativity, the inverse axiom, or the identity axiom. Inverses do the cancelling. The single tool that erases the a is a^ -1 . No inverse, no cancellation. Left, not right. The a sits on the left of b and c , so we hit it with a^ -1 on the left . Right-multiplying gives the conjugate aba^ -1 and never isolates b — that move proves the right cancellation law instead. It fails without inverses. Drop the inverse axiom and cancellation can collapse. In under multiplication, , yet — because 2 is a zero divisor ( ) with no inverse mod 6 . Cancellation is exactly the "no nonzero zero divisors" condition that defines an integral domain.

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