Verified Solution • 2025–26 Session moderate

QJEEM22Ma9825: JEE Main 2022 Mathematics - Class 11 JEEM22Ma9825 Solution: Let , { , } be such that p q ((p q) r) is a tautology. Then (p…

Let Δ\Delta , ∇\nabla ∈\in { ∧\wedge , ∨\vee } be such that p ∇\nabla q ⇒\Rightarrow ((p Δ\Delta q) ∇\nabla r) is a tautology. Then (p ∇\nabla q) Δ\Delta r is logically equivalent to :

Multiple Choice Options

  • A (p Δ\Delta r) ∨\vee q
  • B (p Δ\Delta r) ∧\wedge q
  • C (p ∧\wedge r) Δ\Delta q
  • D (p ∇\nabla r) ∧\wedge q

Step-by-Step Verified Solution

Case-I

If ∇\nabla is same as ∧\wedge

Then (p∧q)⇒((pΔq)∧r)(p \wedge q) \Rightarrow((p \Delta q) \wedge r)

is equivalent to ∼(p∧q)∨\sim(p \wedge q) \vee ((pΔq)∧r)((p \Delta q) \wedge r)

is equivalent to (∼(p∧q)∨(pΔq))∧(∼(p∧(\sim(p \wedge q) \vee(p \Delta q)) \wedge(\sim(p \wedge q)∨r)q) \vee r)

Which cannot be a tautology

For both Δ\Delta (i.e. ∨\vee or ∧\wedge )

Case-II

If ∇\nabla is same as vv

Then (p∨q)⇒((pΔq)∨r)(p \vee q) \Rightarrow((p \Delta q) \vee r) is equivalent to ∼(p∨q)∨(pΔq)∨r\sim(p \vee q) \vee(p \Delta q) \vee r

which can be a tautology if Δ\Delta is also same as ∨\vee .

Hence both Δ\Delta and ∇\nabla are same as vv .

Now (p∇q)Δr(p \nabla q) \Delta r is equivalent to (p∨q∨r)(p \vee q \vee r) .

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