BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 3

Question 1

Which rule of inference allows you to conclude Q from the premises P and P ⟹ Q?

Accepted Answer

Modus ponens

Answer

Modus ponens states: given P and P ⟹ Q, we may conclude Q directly.

Question 2

In mathematical logic, a formula is called a tautology if:

Accepted Answer

It is true under every interpretation

Answer

A tautology is a propositional formula that evaluates to true for all possible truth-value assignments to its variables.

Question 3

Which of the following correctly describes the strong form of mathematical induction?

Accepted Answer

Prove P(1) and that P(1) ∧ … ∧ P(k) ⟹ P(k+1)

Answer

Strong induction assumes all of P(1),…,P(k) are true (not just P(k)) to conclude P(k+1).

Question 4

A function f: A → B is injective (one-to-one) if:

Accepted Answer

f(a₁) = f(a₂) implies a₁ = a₂

Answer

Injectivity means distinct inputs produce distinct outputs: f(a₁) = f(a₂) ⟹ a₁ = a₂.

Question 5

De Morgan's law for quantifiers states that ¬(∀x P(x)) is equivalent to:

Accepted Answer

∃x ¬P(x)

Answer

Negating a universal quantifier yields an existential: ¬(∀x P(x)) ≡ ∃x ¬P(x).

BMath Bachelor of Mathematics Practice Test