🧠 logic

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. **State the problem:** We have two statements: $p$: "The shop is open."

1. The problem involves understanding the logical implications between quantified statements: $\forall x, P(x) \to \exists x, Q(x)$ and $\exists x, P(x) \to Q(x)$.\n\n2. Recall the

1. Let's first understand the problem: We want to create a counterexample for the implication $A \to B$. This means we want a case where $A$ is true but $B$ is false. 2. The implic

1. **Stating the problem:** We are given two logical statements: $A: (\forall x, P(x)) \to (\exists x, Q(x))$ and $B: (\exists x, P(x)) \to Q(x)$. We want to determine if $A \to B$

1. **Problem statement:** We need to determine for each pair of expressions A and B whether the implication $A \to B$ is a tautology and whether $B \to A$ is a tautology.

1. **Stating the problem:** We want to determine if the implication $A \to B$ is a tautology, where: - $A = \exists x \forall y, P(x) \to Q(y)$

1. **Problem statement:** Determine the truth value of each proposition from part II of Exercise 1. 2. **Recall:** The propositions involve quantifiers over real numbers $\mathbb{R

1. **Problem statement:** Determine the truth value of each proposition in Exercise 1, Part II, and justify the answer. 2. **Recall:** The propositions involve quantifiers \(\foral

1. **Problem:** Negate the proposition $\forall x \in \mathbb{R}, \exists y \in \mathbb{R} : 2x + y > 3$. **Step 1:** Recall the negation rules for quantifiers:

1. **State the problem:** Show that $P \leftrightarrow Q$ is equivalent to $\neg P \leftrightarrow \neg Q$ using both truth tables and laws of logic. 2. **Truth Table Method:**

1. **State the problem:** We analyze the logical forms of the statements using the variables:

1. **Stating the problem:** We need to classify each sentence as either a declarative sentence (kalimat deklaratif), an expression with meaning but not declarative, or a sequence o

1. **Stating the problem:** We need to classify each sentence as either a declarative sentence (kalimat deklaratif) which has a definite truth value (true or false), a meaningful e

1. **Problem 1:** Use De Morgan’s Laws to simplify and describe the solution set for: $$\sim(x < -2 \text{ or } x \geq 5)$$

1. Statement of the problem. Problem: Decide whether a given propositional formula is a tautology.

1. **State the problem:** Determine whether the logical formula $$((p \to n2) \lor (p \to q))$$ is a tautology, contingency, or contradiction. 2. **Recall definitions:**

1. You asked for a truth table, which is a tool used in logic and computer science to show all possible truth values of a logical expression. 2. However, you did not provide a spec

1. **Stating the problem:** Determine whether the logical statement $$(P \to Q) \lor (P \to Q)$$ is a tautology, contingency, or contradiction. 2. **Recall definitions:**

1. Logic puzzles involve reasoning to arrive at a conclusion based on given premises or clues. 2. They often require identifying patterns, making deductions, or eliminating possibi

1. **State the problem:** We want to verify the logical argument: "If a number is even, then it is divisible by 2. The number is not divisible by 2. Therefore, the number is not ev