Why isn't the answer to TMUA-P2-ARG2-008 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ARG2-008 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ARG2-008?

No — calculators are banned in the TMUA. TMUA-P2-ARG2-008 is set for a no-calculator target of about 280 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-ARG2-008?

TMUA-P2-ARG2-008 is Paper P2 · Necessary and sufficient (specification code ARG2), level 3. Drill more of the same topic from this page.

TMUA-P2-ARG2-008 — Paper P2 · Necessary and sufficient · Level 3

P2ARG2Level 3~280s target

Spec: ARG2 · No calculator · ~280s

A puzzle book gives two clues about three statements

A

B

and

C

concerning a hidden integer: clue one says that

A

is sufficient for

B

, and clue two says that

B

is necessary for

C

. The book then offers five verdicts about what must follow for

A

and

C

regardless of the hidden integer involved. Three friends argue that a chain can be completed, but the editor insists one particular verdict is the only safe one. Which verdict is logically valid?

Options

$A$ is sufficient for $C$ : whenever $A$ holds, $C$ holds.
$C$ is sufficient for $A$ : whenever $C$ holds, $A$ holds.
$A$ is necessary for $C$ .
$A$ and $C$ are equivalent.
No conclusion relating $A$ and $C$ follows from the two clues alone.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Hint (check trope)

No written hint yet — check the trope: necessary-vs-sufficient-swap, proof-direction-reversed. Name the trap first, work the stem by hand, then open the solution below.

Show solution

The clues give

A \Rightarrow B

and

C \Rightarrow B

: both arrows point into

B

, forming no chain from

A

C

in either direction (e.g.

A

true,

B

true,

C

false is consistent, as is

C

true,

B

true,

A

false). Trap: chaining the arrows head-to-head as if

B

being necessary for

C

meant

B \Rightarrow C

; it means the reverse.

Keep practising

Same topic · same level
P2ARG2Level 3
TMUA-P2-ARG2-014
Three statements about an integer n are given: A is “ n is a multiple of 3 ”, B is “ n is a multiple of 12 ”, and C is “…
Necessary and sufficient
Open TMUA-P2-ARG2-014 →
Same trap: proof-direction-reversed
P2ARG1Level 2
TMUA-P2-ARG1-004
Let P and Q be statements. The claim “ P if and only if Q ” (written P iff Q ) asserts which of the following?
Statements and connectives
Open TMUA-P2-ARG1-004 →
More in this topic
P2ARG2Level 3
TMUA-P2-ARG2-015
A driving school states that passing the theory test is necessary but not sufficient for a full licence, because candida…
Necessary and sufficient
Open TMUA-P2-ARG2-015 →

Browse more

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Options

A

is sufficient for

C

: whenever

A

holds,

C

holds.

C

is sufficient for

A

: whenever

C

holds,

A

holds.

A

is necessary for

C

A

and

C

are equivalent.

No conclusion relating

A

and

C

follows from the two clues alone.

Letters only — the answer key is hidden. Work it out, then check the solution below.

Keep practising

Same topic · same level

P2ARG2Level 3

TMUA-P2-ARG2-014

Three statements about an integer n are given: A is “ n is a multiple of 3 ”, B is “ n is a multiple of 12 ”, and C is “…

Necessary and sufficient

Open TMUA-P2-ARG2-014 →

Same trap: proof-direction-reversed

P2ARG1Level 2

TMUA-P2-ARG1-004

Let P and Q be statements. The claim “ P if and only if Q ” (written P iff Q ) asserts which of the following?

Statements and connectives

Open TMUA-P2-ARG1-004 →

TMUA-P2-ARG2-008 — Paper P2 · Necessary and sufficient · Level 3

Options

Keep practising

Master Necessary and sufficient

TMUA-P2-ARG2-008 — Paper P2 · Necessary and sufficient · Level 3

Options

Keep practising

Master Necessary and sufficient