Direct Proof: Sum of Two Even Numbers

Prove that the sum of any two even numbers is even.

Define what even means: A number $n$ is even if $n = 2k$ for some integer $k$ = Definition established

Let the two even numbers be $a$ and $b$: By definition: $a = 2m$ and $b = 2n$ where $m, n$ are integers = Express in terms of definition

Add the two numbers: $a + b = 2m + 2n = 2(m + n)$ = Factor out 2

Identify the form: $2(m + n)$ is $2 \times (\text{integer})$, which is the definition of even = Matches even definition

State conclusion: Therefore, $a + b$ is even = QED

Answer: The sum of two even numbers is always even because $2m + 2n = 2(m+n)$, which is 2 times an integer.

Proof by Contradiction: Square Root of 2 is Irrational

Prove that $\sqrt{2}$ is irrational.

Assume the opposite: Assume $\sqrt{2}$ is rational, so $\sqrt{2} = \frac{a}{b}$ where $a, b$ are integers with no common factors = Start with contradiction assumption

Square both sides: $2 = \frac{a^2}{b^2}$, so $a^2 = 2b^2$ = $a^2$ is even

Deduce $a$ is even: If $a^2$ is even, then $a$ must be even (if $a$ were odd, $a^2$ would be odd) = $a = 2k$ for some integer $k$

Substitute back: $(2k)^2 = 2b^2$, so $4k^2 = 2b^2$, thus $b^2 = 2k^2$ = $b^2$ is even

Deduce $b$ is even: By same logic, $b$ must be even = Both $a$ and $b$ are even

Identify contradiction: But we said $a$ and $b$ have no common factors. If both are even, they share factor 2! = CONTRADICTION

Conclude: Our assumption was false, so $\sqrt{2}$ is irrational = QED

Answer: $\sqrt{2}$ is irrational because assuming it is rational leads to a logical contradiction.

Direct Proof: Product of Odd Numbers

Prove that the product of two odd numbers is odd.

Define what odd means: A number $n$ is odd if $n = 2k + 1$ for some integer $k$ = Definition established

Let the two odd numbers be $a$ and $b$: $a = 2m + 1$ and $b = 2n + 1$ where $m, n$ are integers = Express using definition

Multiply the numbers: $a \times b = (2m + 1)(2n + 1)$ = Set up multiplication

Expand: $= 4mn + 2m + 2n + 1 = 2(2mn + m + n) + 1$ = Factor out 2

Identify the form: $2(2mn + m + n) + 1$ is $2 \times (\text{integer}) + 1$, which is odd = Matches odd definition

State conclusion: Therefore, $a \times b$ is odd = QED

Answer: The product of two odd numbers is always odd because $(2m+1)(2n+1) = 2(2mn+m+n) + 1$.

Mistake: Using specific examples instead of general proof

Why: Showing that $2 + 4 = 6$ is even only proves that ONE case works, not ALL cases.

Correct: Use variables ($2m + 2n$) to represent ANY two even numbers.

Mistake: Assuming what you want to prove

Why: You cannot use the conclusion in your reasoning - that is circular logic.

Correct: Start only from given information and definitions, then derive the conclusion.

Mistake: Missing justification for steps

Why: Every step must be justified by a definition, axiom, or previously proven theorem.

Correct: Always state WHY each step follows from the previous one.

Mistake: Confusing implication direction

Why: If $P \Rightarrow Q$, knowing $Q$ does NOT tell us $P$ is true.

Correct: Be careful: converse and inverse of a true statement may be false.

Computer Science: Algorithm Correctness

Software engineers use proofs to verify that programs work correctly for all inputs.

Proving a sorting algorithm always produces a sorted list requires mathematical induction.

Cryptography: Security Proofs

Modern encryption relies on mathematical proofs to guarantee security.

The security of RSA encryption is based on the proven difficulty of factoring large numbers.

Legal Reasoning: Building Arguments

Lawyers construct logical arguments similar to mathematical proofs.

Legal reasoning follows 'If the evidence shows X, and X implies guilt, then the defendant is guilty.'

Q: What is the difference between a theorem and an axiom?

A: An axiom is a statement accepted as true without proof. A theorem is a statement that has been proven using axioms and other theorems.

Q: Why do we need proofs if we can test examples?

A: Testing examples only shows specific cases work. A proof guarantees ALL cases work. For example, we cannot test infinitely many numbers, but a proof covers them all.

Q: What does QED mean?

A: QED stands for 'quod erat demonstrandum' (Latin for 'what was to be demonstrated'). It marks the end of a proof.