A few notes on Mathematics

Mathematical Statements

Definitions and Propositions/Theorems/… normally take the form:

1. Object \(a\) has property \(P\).

   \(3\) is prime.

2. Every object of type \(T\) has property \(P\).

   Every integer \(\ge1\) is prime or can be represented as the product of primes, uniquely.

3. There exist objects of type \(T\) having property \(P\).

   There exist infinite sets with different sizes.

4. If statement \(A\), then statement \(B\).

   If a function \(f\) is both injective and surjective, then \(f\) is bijection.

List some problem solving strategies

• What are you trying to answer?
• Do you understand all notation/terminology?
• Can you draw a table/graph/picture to describe what’s going on?

• Can you identify exactly why you are stuck?
• Simplify the problem/consider simple cases (when \(x = 0, 1, 2,...\); when \(x\) is real, integer,…; limiting cases).

• Make conjectures and assumptions — do they initially break down?
• Which theorems can be used based on the assumptions?

• Patterns? Similar examples?
• What strategies are used for similar problems like this?

• Is it better to work backwards?

• Create a concept map (quantity not quality). Which ideas are worth following?

• Take a break/walk (use the deep mind to solve unconsciously).

Review:

• Check all calculations and logic hold.
• Dimensional analysis: Is the answer dimensionally correct? Does the size/sign seem reasonable?
• Can you check using a different method?