Lesson 1.2: Direct Proof, Contradiction, and Counter-Example

Welcome to Lesson 1.2 of Foundation Further Mathematics! 🎓 In this lesson, we will delve into different proof strategies that will lay the groundwork for your mathematical reasoning.

Learning Objectives

By the end of this lesson, you should be able to:

Construct a direct proof from definitions and known results.
Understand proof by contradiction, illustrated by the irrationality of $\sqrt{2}$.
Disprove a universal claim using a counter-example.
Choose the appropriate proof strategy for various statements.
Write a clear direct proof of an elementary result.

Introduction

Mathematics is built on the foundation of logical reasoning and proof. Creating a valid argument is not just about getting the right answer; it's about showing why an answer is correct. Proofs help to reveal the truth of statements in mathematics.

Let's get started with the three main types of proofs: direct proofs, proof by contradiction, and disproof by counter-example. 🧐

H2: Direct Proof

What is Direct Proof?

A direct proof is the most straightforward way to demonstrate a mathematical statement. We start with known definitions and facts and logically deduce the statement to prove.

Example: The Sum of Two Even Numbers

Let's prove the statement: The sum of any two even integers is even.

Definition of an Even Number: An integer $n$ is even if there exists an integer $k$ such that $n = 2k$.
Let us take two even integers: Let $a$ and $b$ be two even integers.

By definition, there exist integers $m$ and $n$ such that:
$a = 2m$
$b = 2n$

Sum of Two Even Integers:

$$egin{align*}

a + b & = 2m + 2n \

$ & = 2(m + n)$

$\end{align*}$$$

Conclusion: Since $m + n$ is an integer, $a + b$ can be expressed as $2$ times an integer. Thus, $a + b$ is even. ✅

Summary

The above steps illustrate how we can build a logical argument to prove something directly.

H2: Proof by Contradiction

What is Proof by Contradiction?

Proof by contradiction involves assuming the opposite of what you want to prove, then showing that this assumption leads to a contradiction.

Example: Irrationality of $\sqrt{2}$

We will prove that $\sqrt{2}$ is irrational.

Assume the opposite: Assume that $\sqrt{2}$ is rational, which means there exist coprime integers $p$ and $q$ (i.e., their greatest common divisor is 1) such that:

$$\sqrt{2} = \frac{p}{q}$$

Squaring both sides:

$$2 = \frac{p^2}{q^2}$$

Rearranging gives:

$$p^2 = 2q^2$$

Observations:

This implies that $p^2$ is even, which means that $p$ must also be even (since the square of an odd number is odd).
Therefore, we can write $p = 2k$ for some integer $k$.

Substituting back in:

Thus, $p^2 = (2k)^2 = 4k^2$,
Substituting into $p^2 = 2q^2$ gives $4k^2 = 2q^2$ or $q^2 = 2k^2$.

Conclusion:

This means $q^2$ is even, hence $q$ is also even.
If both $p$ and $q$ are even, they share $2$ as a common divisor. This contradicts our starting assumption that they are coprime. Hence, $\sqrt{2}$ is irrational. ❌

H2: Counter-Example

What is a Counter-Example?

A counter-example is an example that disproves a universal statement. If we can find a single case where the statement does not hold, we have effectively disproven it.

Example: The Claim All Cats are Black

Claim: All cats are black.

To disprove this statement, consider the cat named Whiskers, which is orange. 😺
Whiskers serves as a counter-example to the claim that all cats are black.
Thus, the statement is false because it does not hold in all cases! 🚫

Choosing the Right Strategy

As you approach a proof, consider which strategy will be most effective. If you can directly prove it, do so. If it seems complex, consider proof by contradiction. If it's a universal claim, look for a counter-example. 🧠

Conclusion

In this lesson, we have explored three key methods of proof: direct proof, proof by contradiction, and counter-example. Developing the ability to construct and understand these proofs is crucial for your success in mathematics. Remember, the goal is not just to find the answer but to understand the reasoning behind it.

H1: Study Notes

Direct proofs rely on definitions and known results.
Proof by contradiction requires assuming the opposite and arriving at a contradiction.
Counter-examples disprove universal claims by showing one instance where the claim is false.
Choose your proof strategy according to the statement you are trying to prove or disprove.
Clarity and logical structure are key elements in writing proofs.