### 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.