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.