CC 103
Tutorial 1 & 2
1. Construct a truth table for each of these compound propositions
i.
ii.
iii.
iv.
v.
𝑝 ∧ ¬𝑝
(𝑝 ∨ ¬𝑞) → 𝑞
(𝑝 → 𝑞) ↔ (¬𝑞 → ¬𝑝)
𝑝⨁𝑝
𝑝⨁ ¬𝑞
2. Construct a truth table for (𝑝 ↔ 𝑞) ↔ (𝑟 ↔ 𝑠).
3. Use truth tables to verify these equivalences
i.
ii.
iii.
iv.
𝑝∧𝑇 ≡𝑝
𝑝∧𝐹 ≡𝐹
𝑝∨𝑝 ≡𝑝
𝑝 ∨𝐹 ≡𝑝
4. Show that 𝑝 → 𝑞 𝑎𝑛𝑑 − 𝑝 ∨ 𝑞 are logically equivalent.
5. Show that 𝑝 ∨ (𝑞 ∧ 𝑟) 𝑎𝑛𝑑 (𝑝 ∨ 𝑞) ∧ (𝑝 ∨ 𝑟) are logically equivalent.
6. Let p be the statement “Maria learns discrete mathematics” and q the statement
“Maria will find a good job”.
Express the statement: p→q
7. What are the contrapositive, the converse and the inverse of the conditional statement
“The hometeam wins whenever it is raining”.
8. Let p, q, r be the propositions:
p: You have the flu
q: You miss the final examination
r: You pass the course
Express each of these propositions in English:
𝑝→𝑞
¬𝑞 ↔ 𝑟
𝑞 → ¬𝑟
9. What are the negations of the statements ∀ 𝑥 (𝑥 2 > 𝑥) and ∃ 𝑥 (𝑥 2 = 2)?
10. Let 𝑃(𝑥) denote the statement "x ≤ 4. ". What are these truth values?
i.
𝑃(0)
ii. 𝑃(4)
iii. 𝑃(6)
11. State the value of 𝑥 after the statement 𝒊𝒇 𝑷(𝒙) 𝒕𝒉𝒆𝒏 𝒙 ≔ 𝟏 is executed, where 𝑷(𝒙)
is the statement "𝒙 > 𝟏" if the value of x when this statement is reached is
i.
𝑥=0
ii. 𝑥 =1
iii. 𝑥 = 2
12. Translate each of these statements into logical expressions using predicates,
quantifiers and logical connectives.
i.
No one if perfect.
ii.
All your friends are perfect.