BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 5

Question 1

Which of the following correctly expresses the negation of 'P if and only if Q'?

Accepted Answer

(P ∧ ¬Q) ∨ (¬P ∧ Q)

Answer

¬(P ↔ Q) ≡ (P ∧ ¬Q) ∨ (¬P ∧ Q), meaning exactly one of P, Q is true.

Question 2

A set S is said to be closed under an operation * if:

Accepted Answer

Applying * to elements of S always yields an element of S

Answer

Closure means for all a, b ∈ S, the result a * b is also in S, keeping results within the set.

Question 3

In proof by contrapositive, to prove P ⟹ Q one instead proves:

Accepted Answer

¬Q ⟹ ¬P

Answer

Since P ⟹ Q is logically equivalent to its contrapositive ¬Q ⟹ ¬P, proving the latter establishes the original implication.

Question 4

Which of the following is NOT a valid rule of inference in classical propositional logic?

Accepted Answer

Affirming the consequent: Q, P ⟹ Q ∴ P

Answer

Affirming the consequent is a formal fallacy; knowing Q is true and P ⟹ Q does not allow us to conclude P.

Question 5

The Well-Ordering Principle and Mathematical Induction over ℕ are:

Accepted Answer

Logically equivalent

Answer

The Well-Ordering Principle (every non-empty subset of ℕ has a minimum) and the Principle of Mathematical Induction are logically equivalent; each can be derived from the other.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 5