BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 4

Question 1

Which of the following is an example of a vacuously true statement?

Accepted Answer

All members of the empty set satisfy every property

Answer

A universally quantified statement over an empty domain (or empty set) is vacuously true because there are no elements to serve as counterexamples.

Question 2

What is the principal difference between an axiom and a theorem?

Accepted Answer

Axioms are assumed without proof; theorems are derived by proof

Answer

Axioms are foundational statements accepted without proof, while theorems are statements derived from axioms and prior results through logical deduction.

Question 3

Which of the following proof strategies is best suited to show that a mathematical object with a certain property does NOT exist?

Accepted Answer

Proof by contradiction

Answer

To show non-existence, assume the object exists and derive a contradiction, thereby disproving its existence.

Question 4

In set theory, the symmetric difference A △ B is defined as:

Accepted Answer

(A ∪ B) \ (A ∩ B)

Answer

The symmetric difference A △ B consists of elements in A or B but not in both, i.e., (A ∪ B) \ (A ∩ B).

Question 5

Gödel's first incompleteness theorem states that any consistent formal system strong enough to express arithmetic:

Accepted Answer

Contains unprovable truths

Answer

Gödel showed every sufficiently powerful consistent formal system contains true statements that cannot be proved within the system.

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Practice Test

BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 4