Question:
The logical statement \(\sim [(p\rightarrow q)\leftrightarrow (p\wedge \sim q)]\) is
The logical statement \(\sim [(p\rightarrow q)\leftrightarrow (p\wedge \sim q)]\) is
Show Hint
Use the fact that the negation of an implication is the conjunction of the antecedent with the negated consequent.
Updated On: Nov 12, 2026
- a fallacy
- equivalent to \(p\wedge q\)
- equivalent to \(p\vee q\)
- a tautology
Show Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
The negation of an implication is \(\sim(p\rightarrow q)\equiv p\wedge\sim q\). So \(p\rightarrow q\) and \(p\wedge\sim q\) always have opposite truth values.
Step 2: Biconditional
A biconditional is true only when both sides have the same truth value. Here the two sides always differ, so \((p\rightarrow q)\leftrightarrow(p\wedge\sim q)\) is always false, i.e. a contradiction.
Step 3: Negate
The negation of a contradiction is always true, so \(\sim[(p\rightarrow q)\leftrightarrow(p\wedge\sim q)]\) is a tautology.
Final Answer:
The given statement is a tautology. \[ \boxed{\text{(D) }\text{a tautology}} \]
The negation of an implication is \(\sim(p\rightarrow q)\equiv p\wedge\sim q\). So \(p\rightarrow q\) and \(p\wedge\sim q\) always have opposite truth values.
Step 2: Biconditional
A biconditional is true only when both sides have the same truth value. Here the two sides always differ, so \((p\rightarrow q)\leftrightarrow(p\wedge\sim q)\) is always false, i.e. a contradiction.
Step 3: Negate
The negation of a contradiction is always true, so \(\sim[(p\rightarrow q)\leftrightarrow(p\wedge\sim q)]\) is a tautology.
Final Answer:
The given statement is a tautology. \[ \boxed{\text{(D) }\text{a tautology}} \]
Was this answer helpful?
0
0
Top MHT CET Mathematics Questions
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- If $\int\frac{f\left(x\right)}{log \left(sin\,x\right)}dx = log\left[log\,sin\,x\right]+c$ then $f\left(x\right)=$
- If $\int\limits^{K}_0 \frac{dx}{2 + 18 x^2} = \frac{\pi}{24}$, then the value of K is
- If $\int^{\pi/2}_{0} \log\cos x dx =\frac{\pi}{2} \log\left(\frac{1}{2}\right)$ then $ \int^{\pi/2}_{0} \log\sec x dx = $
- The point on the curve $y = \sqrt{x - 1}$ where the tangent is perpendicular to the line $2x + y - 5 = 0 $ is
View More Questions
Top MHT CET Mathematical Logic Questions
- If \((\sim p \wedge q)\rightarrow r\) is false, then the truth values of \(p,q,r\) respectively are
- If the symbolic form of the switching circuit is \( \sim p \vee ( p \wedge \sim q) \vee q \), then the current flows through the circuit only if:
- The verbal statement of the same meaning, of the statement 'If the grass is green then it rains in July' is
- Write the statement in symbolic form 'Sandeep neither likes tea nor coffee but enjoys a soft drink'. Where \( p \): Sandeep likes tea, \( q \): Sandeep likes coffee, \( r \): Sandeep enjoys a soft drink
- The logical expression \( [p \wedge (q \vee r)] \vee [\neg r \wedge \neg q \wedge p] \) is equivalent to
View More Questions
Top MHT CET Questions
- During r- DNA technology, which one of the following enzymes is used for cleaving DNA molecule ?
- In non uniform circular motion, the ratio of tangential to radial acceleration is (r = radius of circle, $v =$ speed of the particle, $\alpha =$ angular acceleration)
- The temperature of $32^{\circ}C$ is equivalent to
- The line $5x + y - 1 = 0$ coincides with one of the lines given by $5x^2 + xy - kx - 2y + 2 = 0 $ then the value of k is
- The heat of formation of water is $ 260\, kJ $ . How much $ H_2O $ is decomposed by $ 130\, kJ $ of heat ?
View More Questions