🧠 logic

1. **State the problem:** We want to show that the argument form with premises $p_1, p_2, \ldots, p_n$ and conclusion $q \to r$ is valid if the argument form with premises $p_1, p_

1. **Problem statement:** Prove the validity of the argument form $p \Rightarrow q, p, r \Rightarrow q \vdash r$. 2. **Recall the meaning:** An argument form is valid if whenever a

1. **State the problem:** We need to determine if the argument "Some dogs are pointers. Some dogs are spaniels. Therefore, some pointers are spaniels." is valid. 2. **Understand th

1. **State the problem:** Given the premises: 1. $\forall x [P(x) \Rightarrow Q(x)]$

1. **Problem:** Translate the given statements into propositional functions with quantifiers. 2. **Step 1: Define predicates**

1. The problem asks to identify which sentences are propositions and determine their truth values. 2. A proposition is a declarative sentence that is either true or false, but not

1. **Problem:** Find the argument form for the argument: If Socrates is human, then Socrates is mortal.

1. **State the problem:** We are given three hypotheses: - Randy works hard (denote as $P$).

1. **State the problem:** We are given an argument with premises and a conclusion: - Premise 1: If George does not have eight legs, then he is not a spider.

1. **State the problem:** We are given an argument with premises and a conclusion: - Premise 1: If Socrates is human, then Socrates is mortal.

1. **Problem:** Express the negations of the given statements so that all negation symbols immediately precede predicates. 2. **Recall:** The key rules for negating quantifiers and

1. The problem asks to translate the given logical statements into English, assuming the domain for each variable is all real numbers. 2. Statement a) is \(\forall x \exists y (x <

1. The problem asks to translate logical statements involving quantifiers and inequalities into English, assuming the domain of all variables is all real numbers. 2. Statement a) i

1. Let's start by stating the problem: We want to understand the rules for constructing truth tables in logic. 2. A truth table is a tool used in logic to determine the truth value

1. **Problem statement:** Establish the logical equivalence for part (a) of problem 49: $$(\forall x P(x)) \wedge A \equiv \forall x (P(x) \wedge A)$$ where $x$ does not occur free

1. The problem involves understanding the arrangement and values of numbers on T-shirts in two rows. 2. The top row has three T-shirts with numbers 6, 5, and 8 from left to right.

1. The problem asks to express the proposition \(\exists x P(x)\) where the domain of \(x\) is \(\{-2,-1,0,1,2\}\) without quantifiers, using only disjunctions, conjunctions, and n

1. The problem asks to express the proposition \(\exists x P(x)\) over the domain \(\{-2,-1,0,1,2\}\) without quantifiers, using only disjunctions, conjunctions, and negations. 2.

1. The problem asks to express the proposition \(\exists x P(x)\) for the domain \(\{-2,-1,0,1,2\}\) using only disjunctions, conjunctions, and negations. 2. The existential quanti

1. The problem asks to express the proposition \(\exists x P(x)\) where the domain of \(x\) is \{-2, -1, 0, 1, 2\} using only disjunctions, conjunctions, and negations. 2. The exis

1. **Problem statement:** Express the statement "A student in your class has a cat, a dog, and a ferret" using the predicates $C(x)$, $D(x)$, $F(x)$, quantifiers, and logical conne