# A Simple Proof by Contradiction

Use a proof by contradiction to show that the sum of an irrational number and a rational number is irrational (using the definitions of rational and irrational real numbers; cf p85 of the textbook, or your notes).

Theorem: If is an irrational number and is a rational number, then is irrational.

Proof: For a proof by contradiction, assume that the hypotheses are true (i.e., that is irrational and is  rational) but that the conclusion is false, i.e., is not irrational.

That means is rational, and so by the definition of rational numbers, for integers .

Then .  There are two cases for :

(i) is irrational: this contradicts the hypothesis that s is rational.

(ii) is rational: then for integers . But then meaning r is rational.  But that contradicts the hypothesis that r is irrational.

Since we get a contra

# Proof of “The square of any odd integer is also odd.”

Let’s prove the statement “The square of any odd integer is also odd.”

First we restate the theorem in a way that will lead us towards the proof, by introducing a variable into the statement of the theorem to represent “any (given) odd integer”:

Theorem: If is odd, then is also odd.

In order to prove this, we will need the definition of what it means for a integer to be odd:

Definition: An integer is odd if for some integer .

Proof of theorem: Assume is an odd integer.  Therefore, by definition, for some .  Then This shows that is odd, since , where , so satisfies the definition of being odd.

# Getting Started with Python: Trinket.io

We will use the programming language python later in the semester for some programming projects. Python has the advantage of being relatively easy to get started with.

You should do the following:

1. Create an account on trinket.io, which is a browser-based coding environment that is designed for teaching and coursework.
2. Take a look at one of their introductions to python, such as
• https://hourofpython.trinket.io/a-visual-introduction-to-python
• https://docs.trinket.io/getting-started-with-python
3. Since we are going to be talking about (mathematical) sets, get familiar with Python’s sets (e.g., see https://www.geeksforgeeks.org/sets-in-python/)