Question bank

TMUA-P2-PRF5-001

A proof reads: (1) Let $m = 2j$ and $n = 2k$ be even integers. (2) Then $m + n = 2j + 2k = 2(j + k)$. (3) Hence the sum…

TMUA-P2-PRF5-002

A proof that the square of an odd integer is odd reads: (1) Write $n = 2k + 1$. (2) Then $n^2 = 4k^2 + 4k + 1$. (3) ???…

TMUA-P2-PRF5-003

A question asks: “Prove that for an integer $n$, $n$ is even if and only if $n^2$ is even.” Which proof structure is req…

TMUA-P2-PRF5-004

A proof reads: (1) Suppose $n^2$ is odd. (2) If $n$ were even, $n^2$ would be even. (3) So $n$ is not even, i.e. $n$ is…

TMUA-P2-PRF5-005

A revision guide prints a short proof without a theorem statement: suppose $x$ and $y$ are rational, write $x = \frac{a}…

TMUA-P2-PRF5-006

It is known that statement $A$ implies statement $B$, and statement $B$ implies statement $C$. Which conclusion follows…

TMUA-P2-PRF5-007

Let $P$ be “$n$ is divisible by $24$”, $Q$ be “$n$ is divisible by $8$”, and $R$ be “$n$ is divisible by $4$”. Since $24…

TMUA-P2-PRF5-008

A student must prove the equivalence “$n$ is even if and only if $n^2$ is even”. Which plan suffices?

TMUA-P2-PRF5-009

It is given that $p > q$ and $q > r$, where $p$, $q$, $r$ are real numbers. Which conclusion follows by chaining the ine…

TMUA-P2-PRF5-010

Statements $A$ and $B$ are equivalent ($A \iff B$), and $B$ implies $C$ ($B \Rightarrow C$). Which conclusion follows?

TMUA-P2-PRF5-011

Three statements $P$, $Q$, $R$ satisfy $P \Rightarrow Q$, $Q \Rightarrow R$ and $R \Rightarrow P$. A study group offers…

TMUA-P2-PRF5-012

A student establishes “$n$ is odd if and only if $n^3$ is odd” by passing through squares: odd $n$ gives odd $n^2$ (dire…

TMUA-P2-PRF5-013

Let $P$ be “$n$ is a multiple of $48$”, $Q$ be “$n$ is a multiple of $24$”, and $R$ be “$n$ is a multiple of $8$”. It is…

TMUA-P2Y-PRF-001

A short proof reads: (1) Let $n = 2k$ be an even integer. (2) Then $3n = 6k = 2(3k)$. (3) Hence $3n$ is even. What theor…

TMUA-P2Y-PRF-002

To prove the biconditional: for an integer $n$, $n$ is odd if and only if $3n + 1$ is even, which pair of jobs must both…

TMUA-P2Y-PRF-003

Consider the claim: if $n^2$ is a multiple of $5$ then $n$ is a multiple of $5$. Which sentence is its contrapositive, a…

TMUA-P2Y-PRF-004

A student writes: (1) Let $a = 2j + 1$ and $b = 2k + 1$ be odd integers. (2) Then $a + b = 2j + 2k + 2 = 2(j + k + 1)$.…

TMUA-P2Y-PRF-005

Theorem: if $x$ is rational then $x^2$ is rational. A classmate claims this also shows that $x^2$ rational is sufficient…

TMUA-P2Y-PRF-006

A revision card prints a proof with no title: suppose $m$ and $n$ are consecutive integers with $m < n$, so $n = m + 1$.…