🧠 logic

1. The problem is to create a truth table, which is a table used in logic to show all possible truth values of logical expressions. 2. A truth table lists all possible combinations

1. **State the problem:** We are given a conditional statement: "If a number is even, then it is divisible by 2." We are also told the number is not divisible by 2, and we want to

1. **State the problem:** Verify whether the statement $$\sim(\sim q \to p) \to \sim q$$ is a tautology using the laws of logic. 2. **Recall the implication equivalence:** An impli

1. The problem states: If a number is even, then it is divisible by 2. The number is not divisible by 2. Therefore, the number is not even. 2. This is a classic example of a contra

1. **State the problem:** Verify whether the statement $$\sim(\sim q \top) \to \sim q$$ is a tautology using the laws of logic. 2. **Recall important logical laws:**

1. **State the problem:** Verify whether the statement $$\sim(\sim q \to p) \to \sim q$$ is a tautology using the laws of logic. 2. **Recall the implication equivalence:** An impli

1. We are given two logical expressions to analyze: 18) $ (p \wedge q) \vee r $

1. **Problem 1:** Simplify the logical expression $p \to (p \lor q \lor r)$. - The implication $p \to A$ is logically equivalent to $\neg p \lor A$.

1. The problem asks to express given English statements into symbolic logic form using the propositions: - $p$: The student studies math

1. **Problem Statement:** Determine the truth value of the statement $\forall x \exists y (x^2 = y)$ where the domain of $x$ and $y$ is all real numbers. 2. **Understanding the sta

1. The problem asks to find the correct symbolic statement for "Ruth Adams retired and she did not start her concrete business." Let: - $a$ = "Ruth Adams retired"

1. The problem is to understand how to identify a statement in logic and provide examples. 2. A statement is a sentence that is either true or false, but not both.

1. The problem is to simplify the logical expression $\sim p \wedge r \vee q \vee \sim r$. 2. Recall that $\vee$ is OR, $\wedge$ is AND, and $\sim$ is NOT.

1. The problem states: A is taller than B, and B is taller than C. 2. This means the height order from tallest to shortest is: A > B > C.

1. **State the problem:** We are given that $p \leftrightarrow q$ is true, and we need to show that $p \lor \neg q$ is equivalent to $q \lor \neg p$. 2. **Recall the meaning of $p

1. The problem is to negate the proposition: $$\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : x^2 + y^2 = 1$$

1. The question "d must be true??" is ambiguous without context. 2. To determine if a variable or statement "d" must be true, we need the specific problem or logical conditions inv

1. **State the problem:** We are given two statements and two conclusions, and we need to determine which conclusions logically follow from the statements. 2. **Analyze the stateme

1. The problem asks to express "There are exactly 8 planets in our solar system" using predicate logic, where $P(x)$ means "$x$ is a planet in our solar system". 2. To say "there a

1. **Problem statement:** Determine the truth value of the statement $\forall n \exists m (n^2 < m)$ where the domain for all variables is all integers. 2. **Understanding the stat

1. **Problem (a):** Check if the proposition $$1^3 + 2^3 + 3^3 + \dots + n^3 = \left[\frac{n(n-1)}{2}\right]^2$$ is true for all positive integers $n$ based on the given argument.