🧠 logic

1. We are given the universe $U$ as the set of integers, and predicates: - $P(x)$: $x$ is a prime number

1. **State the problem:** We want to show that the statement $\sim r \to s$ logically follows from the premises $p \to r$, $\sim p \to q$, and $q \to s$. 2. **Analyze the premises:

1. **State the problem:** Show that $ (p \to q) \wedge (p \to r) $ and $ p \to (q \wedge r) $ are logically equivalent. 2. **Recall the implication equivalence:**

1. **State the problem:** We are given the conditional statement: "The dog barks if it sees a stranger." We need to express its contrapositive, converse, and inverse. 2. **Identify

1. **State the problem:** Construct the truth table for the compound statement $ (p \to q) \lor (\neg p \land \neg q) $ and determine if it is a tautology, contradiction, or contin

1. **State the problem:** We need to determine if the logical statement $$(p \wedge q) \to p$$ is a tautology. 2. **Recall definitions:**

1. **State the problem:** We have two propositions: - $p$: I complete my homework.

1. **State the problem:** We want to find the logical equivalence of the expression $$(p \Rightarrow q) \wedge (\neg p \vee q)$$. 2. **Recall the implication equivalence:** The imp

1. The problem asks to find the logical equivalence of the expression $ (p \Rightarrow q) \wedge (\neg p \vee q) $.\n\n2. Recall that the implication $ p \Rightarrow q $ is logical

1. **State the problem:** We want to find the logical equivalence of the expression $$(p \Rightarrow q) \wedge (\neg p \vee q).$$ 2. **Recall the implication equivalence:** The imp

1. **State the problem:** Show that the logical expression $R \to \sim[(\sim R \wedge X) \lor \sim(R \lor X)]$ is a tautology. 2. **Rewrite the expression:** The expression is an i

1. **State the problem:** We are given propositions: - $p$: I bought a lottery ticket this week.

1. The problem asks to express the proposition \(\neg p\) in English. 2. Given \(p\): "I bought a lottery ticket this week."

1. **State the problem:** We need to identify which sentences are propositions and determine their truth values. 2. **Definition:** A proposition is a declarative sentence that is

1. **State the problem:** We need to determine which sentences are propositions and find the truth values of those that are propositions. 2. **Definition:** A proposition is a decl

1. The problem asks to express the exclusive disjunction $p@q$ using only negation ($\sim$), conjunction ($\wedge$), and inclusive disjunction ($\vee$). 2. By definition, $p@q$ mea

1. **State the problem:** Construct a truth table for the compound proposition $ (p \lor q) \lor r $. 2. **List all possible truth values for $p$, $q$, and $r$: Since each variable

1. **State the problem:** Construct a truth table for the compound proposition $ (p \lor q) \lor r $. 2. **List all possible truth values for $p$, $q$, and $r$: Since each variable

1. **State the problem:** Construct a truth table for the compound proposition $ (p \lor q) \lor r $. 2. **List all possible truth values for $p$, $q$, and $r$: Since each can be t

1. **Problem Statement:** Construct truth tables for the compound propositions: a) $p \oplus p$

1. The problem asks us to determine the truth value of each biconditional statement. A biconditional "if and only if" statement $p \iff q$ is true if both $p$ and $q$ have the same