🧠 logic

1. The problem asks us to determine the truth values of the statements based on the given predicate $P(x)$. 2. For the first predicate $P(x)$: "$x \leq 4$".

1. The problem asks to determine the truth values of statements involving inequalities and conditions. 2. For the first problem, $P(x)$ is defined as "$x \leq 4$." We evaluate:

1. The problem asks for the truth values of the statement $P(x): x \leq 4$ for different values of $x$. 2. The statement $P(x)$ is true if $x$ is less than or equal to 4, false oth

1. The problem is to create a truth table for the logical disjunction (OR) operation $p \lor q$. 2. The OR operation $p \lor q$ is true if at least one of $p$ or $q$ is true, and f

1. The problem is to write the truth table for the logical disjunction $p \lor q$. 2. The logical disjunction $p \lor q$ is true if at least one of $p$ or $q$ is true, and false on

1. **Stating the problem:** We want to determine when two or more propositions are logically equivalent. 2. **Definition:** Two propositions $P$ and $Q$ are logically equivalent if

1. Let's start by defining the terms. 2. A **tautology** is a logical statement that is always true, no matter what the truth values of its components are.

1. The problem is to understand the statement "a was right and b was wrong" in a mathematical or logical context. 2. Typically, "a" and "b" could represent variables, propositions,

1. **State the problem:** Simplify the logical expression $$((p \to q) \wedge (\neg q \lor r)) \lor (\neg(p \wedge \neg r))$$ step by step using laws of logic. 2. **Recall key logi

1. **Stating the problem:** Determine the nature (tautology, contradiction, or contingency) of the logical statement $$ (\neg p \wedge (p \to q)) \to q $$. 2. **Recall the definiti

1. The problem is to determine which statements are correct and which are wrong. 2. To do this, each statement must be evaluated based on its content and context.

1. The problem asks us to write the contrapositive of the statement: "If $x > 3$, then $x^2 > 9$." 2. The original statement is a conditional statement of the form "If P, then Q,"

1. The problem asks us to write the contrapositive of the statement: "If $x > 3$, then $x^2 > 9$." 2. The original statement is a conditional statement of the form "If P, then Q,"

1. ปัญหา: ตรวจสอบว่าประพจน์ใดต่อไปนี้มีค่าความจริงเป็นจริงในบางกรณีและเป็นเท็จในบางกรณี (คือเป็นประพจน์ที่ไม่ใช่สาระจริงหรือสาระเท็จตลอด) 2. สูตรและกฎสำคัญ:

1. **Stating the problem:** We are given a logical argument involving propositions about neighbors contacting law enforcement, music volume, and the party being shut down. We need

1. **State the problem:** We analyze the logical argument using propositions and rules of inference to evaluate its validity.

1. **State the problem:** We need to analyze the logical argument about the party shutdown using rules of inference and evaluate its validity. 2. **Identify propositions:**

1. **State the problem:** We need to check the validity of the argument given about the party, neighbors, and police contact. 2. **Identify the propositions:**

1. **State the problem:** We need to analyze the logical statements and use rules of inference to conclude whether the party was shut down.

1. **State the problem:** Simplify the logical expression $ (p \lor q) \land (\neg p \lor q) $ using logical laws. 2. **Recall important logical laws:**

1. The problem is to understand the logical expression "A or B or C or D". 2. In logic, the "or" operator (disjunction) means that the entire expression is true if at least one of