Mathematical Reasoning Summary

Real Analysis · Axiom Academy

The rigorous foundations the whole course runs on — sets and their operations, the language of logic, the proof toolkit, functions, relations, and the size of infinity. Sets are the universal language. Every object in higher math — a number system, a function, a relation — is built from sets and the operations on them. Logic makes statements precise. Connectives and the quantifiers turn vague claims into objects you can negate, combine, and prove. A proof is a chain of valid steps. Four standard shapes — direct, contrapositive, contradiction, induction — cover almost everything; the art is choosing the one that fits the statement's structure. Functions and relations organize structure. Injective / surjective / bijective classify maps; equivalence relations partition a set; orders rank its elements. Infinity has sizes. are all countable, yet Cantor's diagonal argument shows is strictly larger — there is no biggest infinity. Core Concept Sets & Operations A set is an unordered collection of distinct objects. Set-builder notation carves out the elements of a universe U satisfying a property. Membership and containment are the primitive relations. Union : in A or B . Intersection : in both . Difference : in A but not B . Complement . Symmetric difference — in exactly one of the two. Watch the empty set : it is a subset of every set, and . Core Concept De Morgan & Identities

This is the written version of the interactive lesson above. See the full Real Analysis course.