Course Info

Instructor: Mukto Akash
Office/Extension: MC 6508 x36655
Email: mukto.akash@uwaterloo.ca
Office Hours:
Tuesdays noon - 2pm
Thursdays 2 - 3pm
Textbook: Reading, Discovering, and Writing Proofs Version 0.5 (0.42-0.5 work)
Can be Acquired: at SCH as courseware

The content below is heavily influenced on the courseware.

Notes on Learn

The Course Notes are titled
– Reading, Discovering, and Writing Proofs

Statement:
If n is an integer, then \(n^2 ≥ n\).

Proof:
Since n is an integer, either n > 1, n = 0 or n < 0.
When n > 1, multiply both sides by n to get \(n^2 > n\).
When n = 0, we have \(n^2\) =0,so \(n^2 = n\).
Finally, when n < 0,we know that \(n^2 >0\), so \(n^2 > n\).
In all cases, n2 ≥ n is satisfied.

A statement is a sentence that has a definite state of being true or false.

ex. 2 + 2 = 4 (true)
3 + 2 < 5 (false)

not a statement: “Is 7 = 5?”

Definitions

Proposition: A mathematical claim that is posed in the form of a statement that needs to be proven true or demonstrated false by a valid argument.

Theorem: A particularly significant proposition.

Lemma: A proposition that is used to help prove the theorem.

Corollary: A proposition that follows a theorem.

Proposition 1:
For every real number x, \(x^2\) + 1 ≥ 2x

Proof: a series of convincing arguments that leaves no doubt that the stated proposition is true.

The Proof:
Suppose x is a real number.
Therefore, x - 1 must be a real number, and hence
\((x-1)^2\) ≥ 0
Expanding the terms on the left gives \(x^2\) - 2x + 1 ≥ 0. Adding 2x to both sides yields \(x^2\) + 1 ≥ 2x.

Axiom: A statement that is assumed to be true.