21-127 Notes

What is math? — perhaps what you get when you observe the real world, notice patterns, and study those patterns.

Abstraction — ignoring details and looking for the rawest assumptions.

• Pure Math - Logic

  Logic

  Based on Proposition and Proof A proposition is a statement to which it makes sense to assign a truth value (either...

  7/4/2025

• Pure Math - Sets

  Sets

  Based on Defining Sets Informal: a set is a collection of objects called elements More formal: a set is a collections of...

  7/4/2025

• Pure Math - Functions

  Functions

  Based on Defining Function What is a function Given sets $X$ and $Y$, a function $f$ from $X$ to $Y$, or $f:...

  7/4/2025

• Pure Math - Induction

  Induction

  Based on Peano's Axioms Let's forget everything we know about $\mathbb{N}$. We can define it again using Peano's axioms. It turns...

  7/4/2025

• Pure Math - Number Theory

  Number Theory

  Based on Division Theorem The division theorem states that for integers $a, b \in \mathbb{Z}$ with $b \neq 0$, there exists...

  7/4/2025

• Pure Math - Relations

  Relations

  Based on Defining Relation A relation applies to a set and talks about how elements in that set are related. Formally,...

  7/4/2025

• Pure Math - Counting

  Counting

  Based on We're talking about counting how many things are in a set. Not to be confused with counting with...

  7/4/2025

21-127 Notes Omnibus