Proof by deduction, exhaustion and counter-example Quiz

1. What is the sum of the two consecutive odd numbers $2k+1$ and $2k+3$?

• A. $4k+3$
• B. $4k+4$
• C. $2k+3$
• D. $4k+2$

2. Which value of $n$ is a counter-example to 'if $n$ is prime then $n+2$ is prime'?

• A. $n=3$
• B. $n=5$
• C. $n=7$
• D. $n=11$

3. A proof checks the cases '$n$ even' and '$n$ odd'. Which method is this?

• A. Induction
• B. Counter-example
• C. First principles
• D. Exhaustion

4. What is the minimum value of $x^2+4x+9$ for real $x$?

• A. $9$
• B. $13$
• C. $5$
• D. $-2$

5. Which argument proves that $n^2+n$ is even for all integers $n$?

• A. $n(n+1)$ is a product of consecutive integers, so one factor is even.
• B. It is even for $n=1,2,3$.
• C. $n^2$ and $n$ are both even.
• D. It is divisible by 3.