Take this Quiz: A-Level Maths: Proof | Zestly

Question 1

Which of the following statements regarding proof by exhaustion is correct?

Accepted Answer

It requires checking every possible case within a finite set to confirm a property holds.

Question 2

To disprove the statement 'For all integers $n \ge 0$, $n^2 + n + 41$ is a prime number', which of the following is a valid counter-example?

Accepted Answer

$n = 41$

Question 3

Which of the following are necessary steps in a proof by contradiction for the statement '$\sqrt{2}$ is irrational'?

Accepted Answer

Assume that $\sqrt{2} = \frac{a}{b}$ where $a$ and $b$ are coprime integers.

Question 4

Consider the statement: 'The sum of two odd integers is always even.' Which of the following represents a valid deductive proof?

Answer

Check the sum for $1+1, 3+5, 7+9$ and conclude it holds for all.

Answer

Let the integers be $2m+1$ and $2n+1$. Their sum is $2m+2n+2 = 2(m+n+1)$, which is divisible by 2.

Answer

Assume the sum is odd, then $2m+1+2n+1 = 2k+1$, which leads to $2(m+n)+2 = 2k+1$. Since $2(m+n)+2$ is even and $2k+1$ is odd, this is a contradiction.

Answer

Observe that odd numbers are not divisible by 2, so their sum must be divisible by 2.

Question 5

In Euclid's proof for the infinitude of primes, which logic is used?

Answer

Exhaustion of all possible prime numbers up to a large bound $N$.

Answer

Deduction by showing that the prime counting function $\pi(x)$ grows without bound.

Answer

Contradiction by assuming a finite list of primes exists and constructing a new prime not in that list.

Answer

Induction by showing that if $n$ primes exist, then $n+1$ primes must exist.