Question:
The proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is
The proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is
Show Hint
\(p \rightarrow q\) is logically equivalent to \(\neg p \vee q\).
Updated On: Apr 20, 2026
- a tautology
- a contradiction
- neither tautology nor contradiction
- both tautology and contradiction
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
To determine whether the proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is a tautology, contradiction, or neither, let's analyze it step-by-step using a truth table.
The proposition involves two sub-expressions:
- \(p \rightarrow \neg p\): This is known as the implication which is equivalent to \(\neg p \lor \neg p\).
- \(\neg p \rightarrow p\): This is also an implication which is equivalent to \(p \lor p\).
We determine the truth values of these sub-expressions for all possible values of \(p\):
| \(p\) | \(\neg p\) | \(p \rightarrow \neg p\) | \(\neg p \rightarrow p\) | \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) |
|---|---|---|---|---|
| T | F | F | T | F |
| F | T | T | F | F |
From the truth table, we can observe:
- When \(p\) is True, the entire proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is False.
- When \(p\) is False, the entire proposition \((p \rightarrow \neg p) \wedge (\neg p \rightarrow p)\) is also False.
Since the final column of the truth table indicates that the proposition is False for all possible truth values of \(p\), the proposition is a contradiction. A contradiction is a statement that is always false, regardless of the truth values of its components.
Thus, the correct answer is: the proposition is a contradiction.
Was this answer helpful?
0
0
Top MET Mathematics Questions
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- The equation $3^{3x + 4} = 9^{2x - 2},\ x>0$ has the solution
- A parallelogram is constructed on the vectors $\mathbf{a} = 3\alpha - \beta$, $\mathbf{b} = \alpha + 3\beta$. If $|\alpha| = |\beta| = 2$ and the angle between $\alpha$ and $\beta$ is $\dfrac{\pi}{3}$, then the length of a diagonal of the parallelogram is
- Every group of order 7 is
- Let $U_{n} = 2 + 2^{3} + 2^{5} + \cdots + 2^{2n+1}$ and $V_{n} = 1 + 4 + 4^{2} + \cdots + 4^{n-1}$. Then $\displaystyle\lim_{n \to \infty} \dfrac{U_n}{V_n}$ is equal to
View More Questions
Top MET Mathematical Logic Questions
- If $P(n): 1 + 3 + 5 + \cdots + (2n-1) = n^2$ is
- The negation of the compound proposition $p \vee (\neg p \vee q)$ is
- The statement $P(n) = 9^{n} - 8^{n}$, when divided by 8, always leaves the remainder
- $(p \land q) \lor (\sim p \land \sim q)$ is
- If $P(n)$ is a statement such that $P(3)$ is true. Assuming $P(k)$ is true $\Rightarrow P(k+1)$ is true for all $k \ge 3$, then $P(n)$ is true
View More Questions
Top MET Questions
- A coil of 100 turns and area $2 \times 10^{-2}m^2 $ is pivoted about a vertical diameter in a uniform magnetic field and carries a current of 5A. When the coil is held with its plane in north-south direction, it experiences a couple of 0.33 Nm. When the plane is east-west, the corresponding couple is 0.4 Nm, the value of magnetic induction is [Neglect earth's magnetic field]
- $\int\limits_{0}^{8}| x -5| d x$ is equal to
- Radiation, with wavelength 6561 $?$ falls on a metal surface to produce photoelectrons. The electrons are made to enter a uniform magnetic field of $3 \times 10^{-4} T$. If the radius of the largest circular path followed by the electrons is 10 mm, the work function of the metal is close to :
- In intrinsic semiconductor at room temperature number of electrons and holes are
- Which of the following product is formed in the reaction $ CH_3MgBr {->[(i)CO_2][(ii)H_2O]} ? $
View More Questions