BMath Bachelor of Mathematics Bachelor of Mathematics: Mathematical Proof and Logic 2

Question 1

Which proof technique assumes the negation of the conclusion and derives a contradiction?

Accepted Answer

Proof by contradiction

Answer

Proof by contradiction (reductio ad absurdum) assumes ¬P and derives a contradiction, thereby establishing P.

Question 2

In predicate logic, what does the statement ∀x∃y P(x,y) mean?

Accepted Answer

For all x, there exists a y such that P(x,y) holds

Answer

∀x∃y P(x,y) reads 'for every x, there is some y (possibly depending on x) for which P(x,y) is true.'

Question 3

A relation R on a set A is an equivalence relation if it is:

Accepted Answer

Reflexive, symmetric, and transitive

Answer

An equivalence relation must satisfy reflexivity (aRa), symmetry (aRb ⟹ bRa), and transitivity (aRb ∧ bRc ⟹ aRc).

Question 4

Which of the following is the contrapositive of 'If P then Q'?

Accepted Answer

If ¬Q then ¬P

Answer

The contrapositive of P ⟹ Q is ¬Q ⟹ ¬P, which is logically equivalent to the original.

Question 5

In a well-ordered set, every non-empty subset has:

Accepted Answer

A minimum element

Answer

A well-ordered set is one in which every non-empty subset has a least (minimum) element under the given ordering.

BMath Bachelor of Mathematics Practice Test