Proving the Sum of Two Even Numbers is Even

Prove: If $a$ and $b$ are even integers, then $a + b$ is even.

State the hypothesis: Assume $a$ and $b$ are even integers = Given: $a$ and $b$ are even

Apply the definition of even: An even integer can be written as $2k$ for some integer $k$ = $a = 2m$ and $b = 2n$ for integers $m, n$

Add the two numbers: $a + b = 2m + 2n$ = $a + b = 2m + 2n$

Factor out the common factor: $a + b = 2(m + n)$ = $a + b = 2(m + n)$

Apply the definition of even again: Since $m + n$ is an integer, $2(m + n)$ is even by definition = $a + b$ is even

Answer: Therefore, the sum of two even integers is always even. QED

Proving the Square of an Odd Number is Odd

Prove: If $n$ is an odd integer, then $n^2$ is odd.

State the hypothesis: Assume $n$ is an odd integer = Given: $n$ is odd

Apply the definition of odd: An odd integer can be written as $2k + 1$ for some integer $k$ = $n = 2k + 1$ for some integer $k$

Square the expression: $n^2 = (2k + 1)^2$ = $n^2 = (2k + 1)^2$

Expand using algebra: $n^2 = 4k^2 + 4k + 1$ = $n^2 = 4k^2 + 4k + 1$

Factor to show odd form: $n^2 = 2(2k^2 + 2k) + 1$ = $n^2 = 2(2k^2 + 2k) + 1$

Conclude using definition: This is $2m + 1$ where $m = 2k^2 + 2k$ is an integer = $n^2$ is odd

Answer: Therefore, the square of an odd integer is always odd. QED

Proving a Divisibility Property

Prove: If $n$ is an integer, then $n^3 - n$ is divisible by 3.

Factor the expression: $n^3 - n = n(n^2 - 1)$ = $n^3 - n = n(n^2 - 1)$

Factor further using difference of squares: $n(n^2 - 1) = n(n-1)(n+1)$ = $n^3 - n = (n-1) \cdot n \cdot (n+1)$

Recognize the pattern: $(n-1)$, $n$, $(n+1)$ are three consecutive integers = Product of 3 consecutive integers

Apply number theory fact: Among any 3 consecutive integers, exactly one is divisible by 3 = One factor is divisible by 3

Conclude: Since one factor is divisible by 3, the entire product is divisible by 3 = $n^3 - n$ is divisible by 3

Answer: Therefore, for any integer $n$, the expression $n^3 - n$ is divisible by 3. QED

Mistake: Using specific examples instead of general proof

Why: Showing that $2 + 4 = 6$ (even) proves nothing about ALL even numbers. A proof must work for every possible case.

Correct: Use variables and definitions. Write 'Let $a = 2m$ for some integer $m$' to represent ANY even number.

Mistake: Assuming what you want to prove

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

Correct: Start only from the hypothesis and work forward. Never write 'Since $a + b$ is even...' until you have proven it.

Mistake: Skipping logical steps

Why: Each step must follow logically from previous steps. Gaps in reasoning can hide errors.

Correct: Justify every step with a definition, axiom, or previously proven theorem.

Software Verification

Programmers use direct proof logic to verify that their code works correctly for all inputs.

To prove a sorting algorithm always produces sorted output, we show: if the input is a list, the output has the same elements in non-decreasing order.

Legal Reasoning

Lawyers construct arguments that logically lead from evidence to conclusions.

If the contract states payment is due upon delivery, and delivery occurred on March 1, then payment was due on March 1.

Q: How do I know when to use direct proof versus other methods?

A: Try direct proof first for positive statements like 'If A, then B.' If you get stuck, consider proof by contradiction or contrapositive. Direct proof works well when you can manipulate the hypothesis algebraically.

Q: What does QED mean?

A: QED stands for 'quod erat demonstrandum' (Latin for 'which was to be demonstrated'). It signals the proof is complete. Some mathematicians use a square symbol (tombstone) instead.

Q: How detailed should my steps be?

A: Each step should be justified by a definition, property, or theorem. If you skip steps, you risk hiding errors. When in doubt, include more detail.