Which of the following logical implications are tautology? A. $[\sim p \land (p \lor q)] \rightarrow q$ B. $\sim p \rightarrow (p \rightarrow q)$ C. $[p \land (p \rightarrow q)] \rightarrow q$ D. $\sim (p \rightarrow q) \rightarrow q$ Choose the correct answer from the options given below:
Logical Implications Tautology Analysis
To determine which logical implications are tautologies, we need to check if they are true for all possible truth values of the variables $p$ and $q$. We can use logical equivalences or truth tables. Here, we use logical equivalences for conciseness.
Analyzing Option A: $[\sim p \land (p \lor q)] \rightarrow q$
- Using the distributive law: $[\sim p \land (p \lor q)] \equiv (\sim p \land p) \lor (\sim p \land q)$.
- Since $(\sim p \land p)$ is always False (F), the expression simplifies to $F \lor (\sim p \land q)$, which is equivalent to $(\sim p \land q)$.
- The implication becomes: $(\sim p \land q) \rightarrow q$.
- Using the implication equivalence $(\alpha \rightarrow \beta) \equiv (\sim \alpha \lor \beta)$: $(\sim p \land q) \rightarrow q \equiv \sim (\sim p \land q) \lor q$.
- Applying De Morgan's law: $\sim (\sim p \land q) \equiv (\sim (\sim p) \lor \sim q) \equiv (p \lor \sim q)$.
- The expression is now $(p \lor \sim q) \lor q$.
- Rearranging terms: $p \lor (\sim q \lor q)$.
- Since $(\sim q \lor q)$ is always True (T), the expression becomes $p \lor T$, which is always T (True).
- Conclusion: Option A is a tautology.
Analyzing Option B: $\sim p \rightarrow (p \rightarrow q)$
- Using the implication equivalence $(\alpha \rightarrow \beta) \equiv (\sim \alpha \lor \beta)$ twice: $\sim p \rightarrow (p \rightarrow q) \equiv \sim (\sim p) \lor (\sim p \lor q)$.
- Simplifying $\sim (\sim p)$: $\sim p \rightarrow (p \rightarrow q) \equiv p \lor (\sim p \lor q)$.
- Rearranging terms: $(p \lor \sim p) \lor q$.
- Since $(p \lor \sim p)$ is always True (T), the expression becomes $T \lor q$, which is always T (True).
- Conclusion: Option B is a tautology.
Analyzing Option C: $[p \land (p \rightarrow q)] \rightarrow q$
- First, simplify the term in the square brackets: $p \land (p \rightarrow q)$.
- Using the implication equivalence: $p \land (\sim p \lor q)$.
- Applying the distributive law: $(p \land \sim p) \lor (p \land q)$.
- Since $(p \land \sim p)$ is always False (F), this simplifies to $F \lor (p \land q)$, which is equivalent to $(p \land q)$.
- The implication becomes: $(p \land q) \rightarrow q$.
- Using the implication equivalence: $\sim (p \land q) \lor q$.
- Applying De Morgan's law: $(\sim p \lor \sim q) \lor q$.
- Rearranging terms: $\sim p \lor (\sim q \lor q)$.
- Since $(\sim q \lor q)$ is always True (T), the expression becomes $\sim p \lor T$, which is always T (True).
- Conclusion: Option C is a tautology.
Analyzing Option D: $\sim (p \rightarrow q) \rightarrow q$
- Simplify the antecedent $\sim (p \rightarrow q)$: $\sim (\sim p \lor q)$.
- Applying De Morgan's law: $(p \land \sim q)$.
- The implication becomes: $(p \land \sim q) \rightarrow q$.
- Using the implication equivalence: $\sim (p \land \sim q) \lor q$.
- Applying De Morgan's law: $(\sim p \lor \sim (\sim q)) \lor q \equiv (\sim p \lor q) \lor q$.
- This simplifies to $\sim p \lor q$.
- This expression is not always true. For example, if $p$ is True and $q$ is False, $\sim p \lor q$ evaluates to $F \lor F$, which is False.
- Conclusion: Option D is not a tautology.
Final Determination
Based on the analysis, options A, B, and C are tautologies, while option D is not.
Therefore, the correct choice is the one that includes A, B, and C only.
Important Questions from Propositional Logic
- Assume that a Creative ($C$) person will Succeed ($S$) if the person is also Disciplined ($D$), but will not succeed otherwise. Now, consider the following statements:
(i) $C \land S \Leftrightarrow D$
(ii) $C \Rightarrow (S \Leftrightarrow D)$
(iii) $C \Leftrightarrow ((D \Rightarrow S) \lor \neg S)$
Which of the following options is correct? - Let $x$ and $y$ be two propositions. Which of the following statements is a tautology /are tautologies?
Let $p$ and $q$ be any two propositions. Consider the following propositional statements.
$S_1: p \to q$ $S_2: \neg p \land q$ $S_3: \neg p \lor q$ $S_4 : \neg p \lor \neg q$
where $\land$ denotes conjunction (AND operation), $\lor$ denotes disjunction (OR operation), and $\neg$ denotes negation (NOT operation). Which one of the following options is correct?
(Note: $\equiv$ denotes logical equivalence)- Sentence $X$ is said to entail Sentence $Y$ if whenever $X$ is $TRUE$, $Y$ also must hold $TRUE$.
Which of the following statements is/are correct if $X$ entails $Y$? - Let P denote "She is intelligent" and let Q denote "She is happy."
Given below are the statements:
(a) If She is intelligent, then She is unhappy.
(b) She is neither intelligent nor happy.
(c) It is necessary to be not intelligent in order to be happy.
(d) To be not intelligent is to be unhappy.
Which of the following is correct Propositional expression for the above statements:
