MATH1061 / 7861 Assignment 1 Semester 2/2020
Due Thursday 27 August at 1pm on blackboard.
Marks will be deducted for sloppy working. Clearly state your assumptions and conclusions, and
justify all steps in your work.
The marked questions 4 and 6 are required for MATH7861 students only. However, MATH1061 students are encouraged to try these also!
Q1 Use a truth table to determine whether the following statement is a contradiction, a tautology or neither. If it is a contradiction or a tautology, verify your answer using logical equivalences.
((p → q) ∧ (∼ r → ∼ q) ∧ ∼ (r ∧ p)) ←→ (∼ (p ∨ q) ∨ (∼ p ∧ r))
(10 marks)
Q2 Show that the following argument is valid, by adding steps using the rules of inference and/or logical equivalences. Clearly label which rule you used in each step.
1. p→q
2. ∼(q∨r) 3. s→r
∴∼s
Q3 Let p ̸→ q be the operation defined by the following truth table:
p q p̸→q TTF TFT FTF FFF
Express each of the following statements using only the symbols p q ∼ ̸→ ( ): (a) p∨q
(b) p∧q
Justify your answers, using either logical equivalences or truth tables.
(5 marks)
Q4 [MATH7861 only] Give a convincing argument for why it is not possible to express p ∧ q usingonlythesymbols p q ∼ ↔ ( ).
(5 marks)
(Continued on the following page…)
(5 marks)
MATH1061 / 7861
Assignment 1 Semester 2/2020
Q5 Let P (x), Q(x),
R(x), S(x) and T (x, y) denote the following predicates with domain Z:
P(x): x2 =x, Q(x): x≤0, R(x): x2 =x+1,
S(x): x is even,
T(x,y): (x