Why isn't the answer to TMUA-P2-ARG4-009 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ARG4-009 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ARG4-009?

No — calculators are banned in the TMUA. TMUA-P2-ARG4-009 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-ARG4-009?

TMUA-P2-ARG4-009 is Paper P2 · Negation (specification code ARG4), level 3. Drill more of the same topic from this page.

TMUA-P2-ARG4-009 — Paper P2 · Negation · Level 3

P2ARG4Level 3~270s target

Spec: ARG4 · No calculator · ~270s

A stretch-and-challenge sheet makes the ambitious claim that for every integer

n

there exists an integer

m

such that

m > n

and

m

is prime, which amounts to saying there are infinitely many primes with no largest one. The teacher asks the class to write down the exact logical negation of this doubly-quantified claim, warning that both the quantifiers and the inner conjunction must be handled. Which sentence is the correct negation?

Options

For every integer $n$ and every integer $m$ , $m \le n$ and $m$ is composite.
There exists an integer $n$ such that every prime number is less than $n$ .
There exists an integer $n$ with no prime numbers at all.
There exists an integer $n$ such that for every integer $m$ with $m > n$ , $m$ is not prime.
For every integer $m$ there exists an integer $n$ with $m \le n$ or $m$ composite.

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. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Negate outside-in:

\text{not }\forall n \exists m (m > n

and

m

prime) becomes

\exists n \forall m

\text{not}(m > n

and

m

prime), i.e. “Some

n

caps all primes: every

m > n

is non-prime”. Trap: negating only the quantifiers or only the conjunction — both flips are needed.

Keep practising

Same topic · same level
P2ARG4Level 3
TMUA-P2-ARG4-010
An analysis primer states the familiar property of non-negative reals in fully quantified form: for every real number x,…
Negation
Open TMUA-P2-ARG4-010 →
Same trap: quantifier-negation-flip
P2ARG3Level 1
TMUA-P2-ARG3-001
What does the statement “For all integers n, n^2 0 ” assert?
Quantifiers
Open TMUA-P2-ARG3-001 →
More in this topic
P2ARG4Level 3
TMUA-P2-ARG4-017
What is the negation of “If P then ( Q or R )”?
Negation
Open TMUA-P2-ARG4-017 →

Browse more

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Options

For every integer

n

and every integer

m

m \le n

and

m

is composite.

There exists an integer

n

such that every prime number is less than

n

There exists an integer

n

with no prime numbers at all.

There exists an integer

n

such that for every integer

m

with

m > n

m

is not prime.

For every integer

m

there exists an integer

n

with

m \le n

m

composite.

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

Keep practising

Same topic · same level

P2ARG4Level 3

TMUA-P2-ARG4-010

An analysis primer states the familiar property of non-negative reals in fully quantified form: for every real number x,…

Negation

Open TMUA-P2-ARG4-010 →

Same trap: quantifier-negation-flip

P2ARG3Level 1

TMUA-P2-ARG3-001

What does the statement “For all integers n, n^2 0 ” assert?

Quantifiers

Open TMUA-P2-ARG3-001 →

TMUA-P2-ARG4-009 — Paper P2 · Negation · Level 3

Options

Keep practising

Master Negation

TMUA-P2-ARG4-009 — Paper P2 · Negation · Level 3

Options

Keep practising

Master Negation