Phase 1 Exam

Mathematical Language & Proof - Phase Exam

15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches

Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.

Q1 Medium

Which statement about the function f: ℝ→ℝ, f(x)=x² is correct?

It is injective

It is surjective onto ℝ

It is neither injective nor surjective onto ℝ

It is a bijection

Q2 Medium

‘A function is integrable only if it is bounded (on a closed interval).’ This tells us:

boundedness is sufficient for integrability

boundedness is necessary for integrability

integrability is sufficient for boundedness

boundedness and integrability are equivalent

Q3 Easy

To disprove the universal claim ‘every continuous function is differentiable’, the correct move is:

a proof by contradiction

a proof by contrapositive

exhibit one counterexample such as |x| at 0

a direct proof

Q4 Easy

An induction argument with a correct inductive step but NO verified base case proves:

the statement for all n

the statement for no n in general

only the base case

the contrapositive

Q5 Medium

In the definition ‘∀ε>0 ∃N: n≥N ⇒ |a_n−L|<ε’, the threshold N is allowed to depend on:

ε only

both n and ε

nothing

Q6 Easy

To prove A = B for two sets, the standard method is to show:

A ⊆ B only

B ⊆ A only

both A ⊆ B and B ⊆ A

that A and B have the same number of elements

Q7 Easy

The contrapositive of ‘if n² is even then n is even’ is:

if n is even then n² is even

if n is odd then n² is odd

if n² is odd then n is odd

if n is not even then n² is even

Q8 Medium

Proof by contradiction of a statement S proceeds by:

assuming S and deriving S again

assuming ¬S and deriving a logical impossibility

assuming the hypothesis and reaching the conclusion

checking finitely many cases

Q9 Medium

Strong induction differs from weak induction in that the inductive hypothesis assumes:

only P(k)

P(k+1) directly

P(j) for all j from the base up to k

nothing

Q10 Medium

Which sequence converges by the Monotone Convergence Theorem?

a_n=(−1)^n

a_n=n

a_n=1−1/n

a_n=sin n

Q11 Medium

A relation that is reflexive, symmetric, and transitive necessarily produces:

a total order

a partition of the set into disjoint classes

a bijection

a Cartesian product

Q12 Hard

Negate: ‘∀ε>0 ∃δ>0 such that P(ε,δ)’.

∀ε>0 ∀δ>0, ¬P

∃ε>0 ∀δ>0, ¬P

∃ε>0 ∃δ>0, ¬P

∀ε>0 ∃δ>0, ¬P

Q13 Medium

Which statement is most naturally attacked by contrapositive?

√2 is irrational

if n² is odd then n is odd

there are infinitely many primes

the empty set is a subset of every set

Q14 Medium

For proving every integer n≥2 has a prime factorization, the natural technique is:

weak induction

strong induction

a single counterexample

direct computation only

Q15 Hard

To prove 1/n² → 0, choosing N works if for n≥N we get 1/n²<ε. A valid choice of N is any integer greater than:

1/ε

1/√ε

√ε

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