Foundations

The building blocks of mathematics.

Logic

The study of valid reasoning.

• Proposition — A statement that is either true or false
• Truth Table — Systematic listing of all truth values
• Logical Connective — AND, OR, NOT, IMPLIES
• Quantifier — FOR ALL, EXISTS

Sets

Collections of objects.

• Set — A collection of distinct objects
• Element — Membership in a set
• Subset — Containment relation
• Union — Combining sets
• Intersection — Common elements
• Complement — Elements not in a set
• Cartesian Product — Pairs from two sets

Relations and Functions

Mappings between sets.

• Relation — A subset of a Cartesian product
• Function — A special kind of relation
• Injection — One-to-one
• Surjection — Onto
• Bijection — One-to-one correspondence

Proof Techniques

Methods for establishing truth.

• Direct Proof — Assume P, prove Q
• Proof by Contradiction — Assume NOT Q, derive contradiction
• Proof by Contrapositive — Prove NOT P implies NOT Q
• Proof by Induction — Base case + inductive step

All pages follow the same structure: Definition → Why It Matters → Properties → Theorem + Proof → Examples.