3 Equality, Order, and Equivalence

The chapter title may look like just a list, but the notions of equality, order, and equivalence are a critically important trio of related but distinct ideas in mathematics. Together they help us define when two things are the “same.” Instruction in these ideas starts on almost the first day of school when students first learn how to count. Students continue study and expand these ideas in their study of mathematics: many of the theorems of mathematics labeled as “Fundamental” are statements of equivalence. These include the Fundamental Theorem of Arithmetic that every integer can be uniquely factored into primes, the Fundamental Theorem of Calculus describing the relationship between antiderivatives and infinite sums, and isomorphism theorems of abstract algebra and homeomorphisms of topology.