3. Explain conditional and biconditional statements with examples. 4. Check whether the following propositions are equivalent. Include a truth table and a few words explaining how... 3. Explain conditional and biconditional statements with examples. 4. Check whether the following propositions are equivalent. Include a truth table and a few words explaining how the truth table supports your answer. i. ¬(p ∧ q) and ¬p ∨ ¬q ii. ¬(p ∨ q) and ¬p ∧ ¬q iii. p ∨ q → r and (¬r → p) ∧ (¬r → q) iv. p ∧ q → p and p ∨ (¬p ∧ q) v. p → q ∨ r, ¬p ∧ r and p → ¬r → q vi. p ↔ q and (¬p ∨ q) ∧ (¬q ∨ p) 5. Explain universal quantification and existential quantification. 6. What is the truth value of the quantification ∀x(x² < 1) where the domain consists of (i) the positive integers greater than or equal to 1, and (ii) the integers x such that −1 < x < 1. 7. Write the negation of the following quantified statements: (i) ∀x P(x) (ii) ∃x P(x)
Understand the Problem
The question involves a series of logical statements and concepts related to propositional logic, truth values, and quantification in mathematics. It seeks explanations, equivalency checks using truth tables, and the negation of certain quantified statements, all of which pertain to formal logic.
