A direct proof uses these lines: let a,b be even; write a=2r,b=2s for integers r,s; then a+b=2(r+s); so a+b is even. Which order lets each line use facts already established?

• Introduce a,b; add the expressions; choose r,s; conclude evenness.
• Introduce a,b; choose r,s; add the expressions; conclude evenness.
• Choose r,s; conclude evenness; introduce a,b; add the expressions.
• Conclude evenness; introduce a,b; add the expressions; choose r,s.

Which opening line announces the structure of the proof?

• Let a and b be odd integers. Their sum is even, and the reader should accept this without a reason.
• We prove directly that two odd integers have an even sum. Let a and b be odd integers.
• We found this proof after trying several examples of odd integers and calculating their sums.

From 'if X then Y' alone, which pair of roles is guaranteed?

• X suffices for Y, and Y is necessary for X
• X is necessary for Y, and Y suffices for X
• X and Y are unrelated

Theo begins: we prove by contradiction that root 2 is irrational. Suppose root 2 is rational. Theo has announced the structure in his first lines.

Only Ana and Bo are students, and only P and Q are tasks. Ana solved only P; Bo solved only Q. Compare 'for each student there is a task they solved' with 'there is a task every student solved'. Which verdict fits?

• The student-first claim holds; the task-first claim fails.
• The student-first claim fails; the task-first claim holds.
• Both claims hold because they require the same task for all.
• Both claims fail because neither student solved both tasks.

Over the integers, which statement is true?

• There is one fixed integer y that is greater than every integer x.
• For every integer x there is an integer y with y > x.
• The two quantifier orders above describe exactly the same requirement.

A proof uses the hypothesis that n is a prime larger than 2 to conclude n is odd. Where should that hypothesis be cited?

• At the deduction: since n is prime and n > 2, it is odd, because an even prime is 2.
• Only at the start of the proof: the reader must infer which later deductions use the prime hypothesis.
• Nowhere in the proof: calling n prime already rules out every even integer, with no need to exclude 2.

An integer m is divisible by 3. A draft uses this hypothesis to write 'm = 3k'. Which completion states the witness guaranteed by divisibility?

• for every real number k
• for some positive integer k
• for some integer k
• for k = m in every case

A proof needs the reading where each student may have solved a different problem. Which rewrite states that order explicitly?

• There is one problem P such that every student S solved P, with P fixed for the whole group.
• There is a student S who solved every problem P, with that student fixed for the whole task list.
• For every student S there is a problem P that S solved.

Negate the claim 'Every reader can check at least one step.' Which sentence states exactly what must happen for that claim to fail?

• There is a reader who can check every proof step.
• There is a reader who cannot check any step.
• Every reader is unable to check any proof step.
• There is a proof step that no reader can check.