2.2 Techniques of Proof: Proof by Contradiction

Objectives:

• Discover proof techniques for basic conditional propositions

• Apply methods of contrapositive and contradiction to several number theory examples

Example 2.2.1.

Show that the system of equations

$x+2y$ $=3$

$2x+4y$ $=5$

has no solution $(x,y)$.

Blank space for you to write your work.

In the previous example you likely used either the method of substitution or elimination to simplify the system of equations and ended up with an untrue statement. This is a contradiction proof!

A Skeleton Version of a Contradiction Proof for Statement $A$.

Guidelines	Examples
Assume $\neg A$.	Assume by way of contradiction (BWOC) that there is a solution $(x,y)$ to the system of equations given above.
Use definitions to rewrite $\neg A$.	Then for the values $x$ and $y$ of this solution, the proposition $x+2y=3$ is true and the proposition $2x+4y=5$ is true.
Use definitions and known results to derive a contradiction (like $Q\wedge \neg Q$).	But multiplying the first equation by 2 yields $2x+4y=6$. Since $2x+4y=5$ and $2x+4y=6$, by substitution we conclude that $5=6$, which is a contradiction.
(Recommended) Confirm that your only assumption is the statement you wish to disprove.	The only thing we assumed was that there was a solution $(x,y)$ to the system of equations given above.
Conclude $B$.	Assuming that there is a solution to this system yields a condition, so we are forced to conclude there is no solution.

Typically contradiction proofs are seen as harder to use and weaker arguments than direct or contrapositive proofs. However, some statements are quite unwieldy to prove using either of these two methods, and a contradiction proof can be much more elegant. Here is what your proofs will be assessed on:

• Clear - your proofs should clearly state your beginning assumptions, your intermediate deductions, and your ending claim.

• Comprehensive - your proofs should prove the statement completely - no “proofs by example”! They should also make use predominantly of content we have discussed in class, not outside material such as readings external to our textbook.

• Concise - if you can simplify your proof after writing it, you should. Proofreading (haha) is a crucial part of the writing process. Did you break your proof into cases when it didn’t need cases? Would a different proof approach ease the burden of proof?

Example 2.2.2.

Prove by contradiction that, if $x>0$, then ${\displaystyle \frac{1}{x}}>0$. Where did you use that $x>0$?

Blank space for you to write your work.

We will not go through too many examples of contradiction proof… in fact, most proofs you may start as contradiction proofs are actually contrapositive proofs in disguise (like the next example)!

We need to give a few definitions before continuing.

Definition 2.2.3.

We define a real number $x$ to be rational if $x={\displaystyle \frac{p}{q}}$ for $p\in \mathbb{Z}$ and $q\in \mathbb{N}$. A real number $x$ is irrational if it is not rational; i.e., it cannot be written in the form$x={\displaystyle \frac{p}{q}}$ for $p\in \mathbb{Z}$ and $q\in \mathbb{N}$.

Example 2.2.4.

Prove or disprove: if $x+y$ is irrational, then $x$ is irrational or $y$ is irrational.

Blank space for you to write your work.

Example 2.2.5.

Prove or disprove: ${\log }_2(3)$ is an irrational number.

Blank space for you to write your work.