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:
P$\rightarrow$$\neg$Q; (b) $\neg$P$\wedge$$\neg$Q; (c) Q$\rightarrow$$\neg$P; (d) $\neg$P$\rightarrow$$\neg$Q
To determine the correct propositional expressions for the provided statements, we need to analyze each statement and match it with its corresponding logical expression.
- Statement (a): "If She is intelligent, then She is unhappy."
- This can be represented as: P \rightarrow \neg Q
- This statement implies that if she is intelligent, then it follows that she is not happy, which matches our conditional logic.
- Statement (b): "She is neither intelligent nor happy."
- This can be represented as: \neg P \wedge \neg Q
- This expression indicates that both conditions are false: she is not intelligent and she is not happy.
- Statement (c): "It is necessary to be not intelligent in order to be happy."
- This can be represented as: Q \rightarrow \neg P
- Here, the statement is rephrased as if she is happy, then she is not intelligent, which matches the logic of the expression.
- Statement (d): "To be not intelligent is to be unhappy."
- This can be represented as: \neg P \rightarrow \neg Q
- This indicates that if she is not intelligent, then it follows that she is not happy.
Thus, the correct propositional expressions for the statements are:
- (a) P \rightarrow \neg Q
- (b) \neg P \wedge \neg Q
- (c) Q \rightarrow \neg P
- (d) \neg P \rightarrow \neg Q
Important Questions from Propositional Logic
- Assume that a Creative ($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)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:
- 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 $Y$?
