Why isn't the answer to TMUA-P2-ARG3-007 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ARG3-007 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ARG3-007?

No — calculators are banned in the TMUA. TMUA-P2-ARG3-007 is set for a no-calculator target of about 270 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-ARG3-007?

TMUA-P2-ARG3-007 is Paper P2 · Quantifiers (specification code ARG3), level 3. Drill more of the same topic from this page.

TMUA-P2-ARG3-007 — Paper P2 · Quantifiers · Level 3

P2ARG3Level 3~270s target

Spec: ARG3 · No calculator · ~270s

A number-theory worksheet asks pupils to compare two claims about even sums, where all letters denote integers: claim one says “There exists an integer

k

such that for every integer

n

, the sum

k + n

is even”, while claim two says “For every integer

n

there exists an integer

k

such that

k + n

is even”. Pupils must decide which claim holds and explain the role of quantifier order in their answer. Which verdict is correct?

Options

Both claims are true, since $k + n$ can always be made even.
Both claims are false, since no choice of $k$ ever works.
Claim two is true (given $n$ , take $k = n$ ) but claim one is false (no single $k$ works for both even and odd $n$ ).
Claim one is true but claim two is false.
The two claims are logically equivalent.

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: quantifier-negation-flip, single-counterexample. Name the trap first, work the stem by hand, then open the solution below.

Show solution

In claim two

k

may depend on

n

(take

k = n

, so

k+n = 2n

is even). In claim one a single

k

must work for all

n

, impossible since

k+n

flips parity with

n

. Trap: assuming the quantifiers commute — nesting order decides whether the witness may depend on the earlier variable.

Keep practising

Same topic · same level
P2ARG3Level 3
TMUA-P2-ARG3-014
Let U be “ x\, y\,(y > x) ” and V be “ y\, x\,(y > x) ” over R. Which is correct?
Quantifiers
Open TMUA-P2-ARG3-014 →
Same trap: single-counterexample
P2ARG1Level 3
TMUA-P2-ARG1-009
A textbook exercise states the following facts about positive integers, where P is “ n is divisible by 6 ” and Q is “ n…
Statements and connectives
Open TMUA-P2-ARG1-009 →
More in this topic
P2ARG3Level 3
TMUA-P2-ARG3-015
What is the negation of “For every real x there is a real y such that for every real z, xyz=0 ”?
Quantifiers
Open TMUA-P2-ARG3-015 →

Browse more

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Options

Both claims are true, since

k + n

can always be made even.

Both claims are false, since no choice of

k

ever works.

Claim two is true (given

n

, take

k = n

) but claim one is false (no single

k

works for both even and odd

n

Claim one is true but claim two is false.

The two claims are logically equivalent.

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

Keep practising

Same topic · same level

P2ARG3Level 3

TMUA-P2-ARG3-014

Let U be “ x\, y\,(y > x) ” and V be “ y\, x\,(y > x) ” over R. Which is correct?

Quantifiers

Open TMUA-P2-ARG3-014 →

Same trap: single-counterexample

P2ARG1Level 3

TMUA-P2-ARG1-009

A textbook exercise states the following facts about positive integers, where P is “ n is divisible by 6 ” and Q is “ n…

Statements and connectives

Open TMUA-P2-ARG1-009 →

TMUA-P2-ARG3-007 — Paper P2 · Quantifiers · Level 3

Options

Keep practising

Master Quantifiers

TMUA-P2-ARG3-007 — Paper P2 · Quantifiers · Level 3

Options

Keep practising

Master Quantifiers