Why isn't the answer to TMUA-P2-ERR1-010 shown upfront?

The options are lettered without marking the correct one so you attempt TMUA-P2-ERR1-010 first — TMUA questions reward spotting the trap before you check the worked solution below.

Do I need a calculator for TMUA-P2-ERR1-010?

No — calculators are banned in the TMUA. TMUA-P2-ERR1-010 is set for a no-calculator target of about 290 seconds; the worked solution shows the fastest by-hand route.

Which TMUA specification point is TMUA-P2-ERR1-010?

TMUA-P2-ERR1-010 is Paper P2 · Identifying errors in proofs (specification code ERR1), level 3. Drill more of the same topic from this page.

TMUA-P2-ERR1-010 — Paper P2 · Identifying errors in proofs · Level 3

P2ERR1Level 3~290s target

Spec: ERR1 · No calculator · ~290s

A flawed lemma in a student journal claims that from the true statement “For every real

x

there exists a real

y

with

y > x

” one may deduce “There exists a real

y

such that for every real

x

y > x

”. The printed justification labels each move: line one restates the premise; line two swaps the quantifiers “since order does not matter”; line three names the resulting

y

as the largest real number. A referee must pinpoint the exact failure. Which report is correct?

Options

Line one misstates the premise, which is false.
Line three is fine but line one is wrong.
Line two is the first fault: $\forall x \exists y$ does not imply $\exists y \forall x$ ; the witness $y$ may depend on $x$ (e.g. $y = x + 1$ ), so no single largest real follows.
All three lines are valid and a largest real exists.
The error is that $y = x + 1$ is not real.

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: line-number-fault, quantifier-negation-flip. Name the trap first, work the stem by hand, then open the solution below.

Show solution

Quantifier order is load-bearing: the premise lets

y

depend on

x

, while the conclusion demands one

y

above all reals — impossible. Trap: treating

\forall

and

\exists

as commuting operators like addition; swapping them strengthens the claim enormously and invalidly.

Keep practising

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TMUA-P2-ERR1-009
A six-line algebra proof submitted for a scholarship problem claims to show that x = 5 follows from the equation (x - 2)…
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Open TMUA-P2-ERR1-009 →
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
P2ERR1Level 3
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A purported proof claims “The product of two irrational numbers is irrational”: (1) Let a and b be irrational. (2) Write…
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Open TMUA-P2-ERR1-015 →

Browse more

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Options

Line one misstates the premise, which is false.

Line three is fine but line one is wrong.

Line two is the first fault:

\forall x \exists y

does not imply

\exists y \forall x

; the witness

y

may depend on

x

(e.g.

y = x + 1

), so no single largest real follows.

All three lines are valid and a largest real exists.

The error is that

y = x + 1

is not real.

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

Keep practising

Same topic · same level

P2ERR1Level 3

TMUA-P2-ERR1-009

A six-line algebra proof submitted for a scholarship problem claims to show that x = 5 follows from the equation (x - 2)…

Identifying errors in proofs

Open TMUA-P2-ERR1-009 →

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-ERR1-010 — Paper P2 · Identifying errors in proofs · Level 3

Options

Keep practising

Master Identifying errors in proofs

TMUA-P2-ERR1-010 — Paper P2 · Identifying errors in proofs · Level 3

Options

Keep practising

Master Identifying errors in proofs