🧠 logic

1. **State the problem:** Determine if each of the following logical statements is a tautology (always true). 2. **Recall:** A tautology is a formula that is true in every possible

1. **State the problem:** We want to prove that the biconditional operation is associative, i.e., $$(p \leftrightarrow q) \leftrightarrow r = p \leftrightarrow (q \leftrightarrow r

1. The problem is to analyze the statement "No car is stolen." 2. This is a universal negative statement, which means it asserts that there are no cars that are stolen.

1. The problem asks if I can solve logic puzzles. 2. Logic puzzles involve reasoning and deduction to find solutions based on given clues.

1. **Stating the problem:** Ashok is looking at a portrait of a man and says, "His mother is the wife of my father's son. Brothers and sisters, I have none." We need to find out wh

1. **Stating the problem:** Amar points to a photograph and says, "I have no brother or sister but that man's father is my father's son." We need to find whose photograph it is and

1. **Problem Statement:** Find the PCNF (Principal Conjunctive Normal Form) and PDNF (Principal Disjunctive Normal Form) of the formula $$ (p \to (q \wedge r)) \wedge (\neg p \to (

1. The problem asks to identify which item does not belong to a group. 2. To solve this, we need to know the items in the group and their characteristics.

1. **Problem:** Construct the truth table for the compound proposition $ (p \wedge q) \to (\neg q \lor p) $. 2. **Formula and rules:** The implication $A \to B$ is false only when

1. The problem asks to identify which proposition has truth value True (T). 2. Let's analyze each option:

1. The problem is to identify the error in the argument that tries to prove: if <span style='color:#1e6fff; font-weight:600; text-shadow:0 0 4px rgba(30,111,255,0.5)'>$$\forall x (

1. **State the problem:** We want to identify the error in the argument that tries to prove: $$\forall x (P(x) \lor Q(x)) \implies \forall x P(x) \lor \forall x Q(x)$$

1. השאלה היא האם הטענה נכונה. 2. כדי לבדוק זאת, יש צורך לדעת מהי הטענה המדוברת.

1. Statement of the problem: We are given the premises $\forall x\, (P(x) \to Q(x))$ and $\neg Q(a)$ for a particular element $a$ in the domain, and we must show $\neg P(a)$.

1. **State the problem:** We want to justify the rule of universal modus tollens, which states that from the premises $\forall x (P(x) \to Q(x))$ and $\neg Q(a)$ for a particular e

1. **Problem statement:** Show that from the premises $\forall x (P(x) \to Q(x))$ and $\neg Q(a)$ for a particular element $a$, we can conclude $\neg P(a)$.

1. **State the problem:** We want to show that the compound proposition $ (p \lor q) \land (\neg p \lor q) \land (p \lor \neg q) \land (\neg p \lor \neg q) $ is not satisfiable usi

1. **State the problem:** We want to show that the compound proposition $ (p \lor q) \land (\neg p \lor q) \land (p \lor \neg q) \land (\neg p \lor \neg q) $ is not satisfiable usi

1. **State the problem:** The argument tries to prove that from $\exists x P(x) \wedge \exists x Q(x)$, it follows that $\exists x (P(x) \wedge Q(x))$. 2. **Recall the logical rule

1. **Problem Statement:** Determine whether the composite statement $$[P \wedge (Q \Rightarrow R)] \Rightarrow [(P \wedge Q) \lor (P \wedge R)]$$ is a tautology. 2. **Recall the de

1. **Problem statement:** Determine the validity of each argument and identify the rule of inference or logical error. 2. **Recall important rules:**