Explain conditional and biconditional statements with examples. Check whether the following propositions are equivalent. Include a truth table and a few words explaining how the tr... Explain conditional and biconditional statements with examples. Check whether the following propositions are equivalent. Include a truth table and a few words explaining how the truth table supports your answer. 1. ¬(p ∧ q) and ¬p ∨ ¬q 2. ¬(p ∨ q) and ¬p ∧ ¬q 3. p ∨ q → r and (p → r) ∧ (q → r) 4. p ∨ q → p and p ∨ ¬q 5. p ∨ q ∨ r, p ∧ ¬q → r and p ∧ ¬r → q 6. p → q and (¬p ∨ q) 7. Explain universal quantification and existential quantification. 8. 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. 9. Write the negation of the following quantified statements: (i) ∀x P(x), (ii) ∃x P(x). Read more

Steps to Solve

1. Setting Up the Truth Table
   First, we will construct a truth table for each proposition from the list provided. Let's denote the propositions as follows:

• $p$: True or False
• $q$: True or False
• $r$: True or False

Each column in the truth table will represent the truth values of the propositions based on every combination of truth values for $p$, $q$, and $r$.

2. Propositions to Evaluate
   We will evaluate the following propositions using logical operations:

• (i) $\neg(p \land q) \lor \neg q$
• (ii) $\neg(p \land q) \land \neg p \land \neg q$
• (iii) $p \lor q \lor (p \rightarrow r) \land (q \rightarrow r)$
• (iv) $p \land q \rightarrow p \land (p \land q)$
• (v) $p \lor (q \lor r) \land (p \land \neg r \rightarrow q)$
• (vi) $p \rightarrow q \land (\neg p \lor q) \land (q \lor p)$

3. Construct the Truth Table for Each Proposition
   Note the results for each row based on every combination of $p$, $q$, and $r$. Each row represents a possible outcome, where True = T and False = F.

4. Comparing the Truth Values
   Once we have the truth table filled out for the propositions, we will compare their final columns to see if they yield the same truth values for each combination of $p$, $q$, and $r$. If the truth values match for all combinations, the propositions are equivalent.

5. Explain Universal and Existential Quantification

• Universal Quantification ($\forall$) asserts that a statement is true for every element in a specified set.
• Existential Quantification ($\exists$) asserts that there is at least one element in a set for which the statement is true.

6. Evaluate Quantification
   For the expression $\forall x (x^2 < 1)$, we analyze it under the specified domains:

• (i) For positive integers $\geq 1$: This quantification is false since no positive integer squared is less than 1.
• (ii) For integers $x$ such that $-1 < x < 1$: This quantification is true for $x=0$.

7. Negate the Quantified Statements
   To negate the statements:

• (i) The negation of $\forall x P(x)$ is $\exists x \neg P(x)$.
• (ii) The negation of $\exists x P(x)$ is $\forall x \neg P(x)$.