Let \(\Delta, \nabla \in\{\Lambda, V\}\) be such that \(( p \rightarrow q ) \Delta( p \nabla q )\) is a tautology. Then
Let \(\Delta, \nabla \in\{\Lambda, V\}\) be such that \(( p \rightarrow q ) \Delta( p \nabla q )\) is a tautology. Then
- $\Delta=\Lambda, \nabla=\Lambda$
- $\Delta=V, \nabla=V$
- $\Delta=V, \nabla=\wedge$
- $\Delta=\Lambda, \nabla=V$
The Correct Option is B
Approach Solution - 1
For the given expression to be a tautology, every possible valuation of \(p\) and \(q\) must make the expression true. Since \(p \to q\) is equivalent to \(\neg p \lor q\), the expression simplifies as:
\( (\neg p \lor q) \land (p \lor q) \)
Using distributive laws:
\( (\neg p \land p) \lor (\neg p \land q) \lor (p \land q) \lor (q \land q) \)
Simplifying further, knowing \(\neg p \land p\) is always false:
\( (\neg p \land q) \lor (p \land q) \lor q = q \)
Hence, for the expression to be a tautology, it must always evaluate to true, which is the case when \(\land\) and \(\lor\) are defined such that the final result of any expression involving these operators is always true.
Approach Solution -2
The correct answer is (B) : \(\Delta=V, \nabla=V\)
Given (p→q)Δ(p∇q)
Option I Δ=∧,∇=∨
Hence, it is tautology.
Option 4Δ=∧,∇=∧
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Concepts Used:
Types of Vectors
In general, vectors are used in Maths and Science and are categorized into 10 different types of vectors such as:-
- Unit Vector - When a vector has a magnitude of 1 unit length, called a Unit Vector.
- Co-Initial Vector - Two or more vectors that have the same initial point are known to be Co-Initial Vectors.
- Coplanar Vector - Vectors that lie either in the same plane or are parallel to the same plane are called Coplanar vectors.
- Equal Vector - When two vectors have equal direction as well as magnitude, they are Equal Vectors, even if the initial point is different for both vectors.
- Negative of a Vector - When two vectors have the same magnitude but have exactly different directions.
- Zero Vector - When a vector has the same starting and ending point and has zero magnitudes is called a zero vector. The starting point needs to coincide with the terminal point. It is denoted by 0. It is also known as the null vector.
- Position Vector - A vector that indicates the location or the position of a point in a plane (three-dimensional Cartesian system) w.r.t. its origin. If A is a reference origin and there’s an arbitrary point B in the plane then AB will be called the position vector of the point.
- Like and Unlike Vectors - Like vectors are the vectors that have the same direction. And unlike vectors are the vectors that have opposite directions.
- Collinear Vector - Vectors either lying in the same line or which are parallel to the same line are Collinear vectors.
- Displacement Vector - The vector AB will be known as a displacement vector if a point is displaced from position B to position A.