🧠 logic

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

1. **Show that $P : p \to (q \to r)$ and $Q : (p \wedge q) \to r$ are logically equivalent.** (a) Using a truth table:

1. **Problem statement:** Show the equivalences involving the XOR operator $p \oplus q$.

1. The problem states that there are multiple options and exactly 2 of them are correct. 2. To analyze this, we need to consider the total number of options and identify which 2 ar

1. The user asks if a given statement is true or false. 2. However, no specific statement or problem is provided to evaluate.

1. The phrase "at least one is wrong" typically refers to a situation where among multiple statements or conditions, one or more are incorrect. 2. To analyze this mathematically, c

1. **Problem:** Given the propositions: - If I am the King, then you are the Queen.

1. **State the problem:** We need to determine if the expression $$(((P \oplus Q) \wedge (Q \lor R)) \to (P \lor R)) \leftrightarrow ((P \to R) \lor (Q \leftrightarrow R))$$ is a t

Problem: Verify that $ (p \vee q) \wedge (p \vee r) $ is logically equivalent to $ p \vee (q \wedge r) $. 1. Approach: We prove the equivalence by showing each direction separately

1. Problem: Determine the validity of each argument. (a) Given: $p \to q$, $q$. Conclusion: $p$.

1. The problem is about understanding the logical conjunction (AND) operation between two propositions $p$ and $q$. 2. The conjunction $p \wedge q$ is true only when both $p$ and $

1. **Stating the problem:** We are given propositions and their logical implications, converses, contrapositives, and inverses. We need to understand these concepts and apply them

1. The problem involves translating English statements about stocks, interest rates, and John's attributes into logical expressions using propositional logic. 2. For the first set:

1. **Problem statement:** Express the given statements using predicates and quantifiers. 2. **Define predicates:**

1. The problem asks to translate logical statements into English, where $R(x)$ means "$x$ is a rabbit" and $H(x)$ means "$x$ hops", with the domain being all animals. 2. For statem

1. **State the problem:** Simplify and analyze the logical expression $$((p \Rightarrow q) \wedge (\neg q)) \Rightarrow \neg q$$. 2. **Recall definitions:**

1. **Problem 1:** Given the predicate $P(x,y): xy = 1$ with $x,y \in \mathbb{R}$, evaluate the truth value of the statement $\forall x \in \mathbb{R}, \exists y \in \mathbb{R}$ suc

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