Proof – Q&A Revision Notes: Direct Proof, Counter-Examples & Divisibility

Thirteen Q&A-style flashcards summarising key definitions, proofs, and counter-examples from the revision notes on direct proof, counter-examples, and divisibility arguments.

What is meant by “proof by deduction” (direct proof)?

It is a method of proving a statement true by starting from accepted facts, definitions and axioms and applying logical reasoning step-by-step until the desired conclusion is reached.

How do you show that the product of two rational numbers is always rational?

Write the rationals as a = p/q and b = r/s with integers p, q, r, s and q, s ≠ 0. Then ab = (p/q)(r/s) = pr/qs, a quotient of two integers with non-zero denominator, hence rational.

What is a disproof by counter-example?

It is a way to refute a universal mathematical statement by exhibiting just one specific example for which the statement fails.

Give a counter-example that disproves: “For any real numbers x > y, we have x² + x > y² + y.”

Let x = 2 and y = −4. Then x > y, but x² + x = 4 + 2 = 6 while y² + y = 16 − 4 = 12, so 6 < 12. The claim is false.

Provide a counter-example for: “If x/y < 1, then x < y.”

Take x = −1 and y = −2. Here x/y = (−1)/(−2) = 0.5 < 1, yet x = −1 > −2 = y, so the statement is false.

Disprove the claim: “If x² + y = y² + x, then x = y.”

Choose x = 2 and y = 1. Then x² + y = 4 + 1 = 5, while y² + x = 1 + 2 = 3; the two sides are unequal, hence x ≠ y, so the claim is false.

How do you prove that n + 36 − (n + 1)² is divisible by 5 for every integer n?

Expand: n + 36 − (n² + 2n + 1) = −n² − n + 35. Reduce each possible n modulo 5: n ≡ 0,1,2,3,4 gives the expression ≡ 0 mod 5 in every case, so the expression is always a multiple of 5.

Why is the difference between any integer and its square always even?

For integer n, n² − n = n(n − 1). The factors are two consecutive integers, one of which is even, so their product – and therefore n² − n – is even.

Is the product of a rational number and an irrational number always irrational?

No. Example: 0 (rational) × √2 (irrational) = 0, which is rational. Therefore the statement is false.

Prove that if A and B are consecutive even integers, then 3A² + 3B² is a multiple of 12.

Let A = 2n and B = 2n + 2. Then 3A² + 3B² = 3(4n²) + 3[4(n² + 2n + 1)] = 12n² + 24n + 12 = 12(n² + 2n + 1), which is clearly divisible by 12.

Does the result in the previous card remain true if A and B are even but not consecutive?

Not necessarily. For example, A = 2 and B = 6 give 3A² + 3B² = 12 + 108 = 120, which you must check for divisibility to see the statement can fail, so the claim is not always true for non-consecutive evens.

Show that if n is divisible by 3, then f(n) = n² − 7 is divisible by 3.

Write n = 3k. Then f(n) = 9k² − 7 ≡ 0 − 7 ≡ −7 ≡ 0 (mod 3) because −7 ≡ 0 (mod 3). Hence f(n) is a multiple of 3.

Prove that if n is not divisible by 3, then f(n) = n² − 7 is still divisible by 3.

If n ≡ 1 (mod 3), then n² ≡ 1, so n² − 7 ≡ 1 − 7 ≡ −6 ≡ 0 (mod 3). If n ≡ 2 (mod 3), then n² ≡ 4 ≡ 1, giving the same remainder. Thus in both cases f(n) is divisible by 3.