Question

Let\( Δ,▽∈{∧,∨} \)
be such that \(p▽q⇒((pΔq)▽r) \)
is a tautology. Then \((p▽q)Δr \)
is logically equivalent to:

• A. \((pΔr)∨q\)
• B. \((pΔr)∧q\)
• C. \((p∧r)Δq\)
• D. \((p▽r)∧q\)

Answer 1

The Correct Option is A

To solve this problem, we need to determine the logical operators \( Δ \) and \( ▽ \) such that the expression \( p ▽ q \Rightarrow ((p Δ q) ▽ r) \) is a tautology. A tautology is a statement that is true under all possible truth assignments.

Let us analyze the expression step-by-step:

1. The implication \( p ▽ q \Rightarrow ((p Δ q) ▽ r) \) can be rewritten using the equivalence of implication: \( A \Rightarrow B \equiv \neg A \vee B \). Therefore, the expression becomes \( \neg (p ▽ q) \vee ((p Δ q) ▽ r) \).
2. Since the expression must be a tautology, it should be true for all possible truth values of \( p, q, \) and \( r \).
3. Consider each possible combination of values for \( Δ \) and \( ▽ \) from \( \{ \land, \lor \} \):(conjunction and disjunction).

Let's analyze the expression with the possible operators:

1. If \( ▽ = \lor \) and \( Δ = \land \):

   The expression becomes \( p \lor q \Rightarrow ((p \land q) \lor r) \).

   The implication \( p \lor q \Rightarrow ((p \land q) \lor r) \) would be: \( \neg (p \lor q) \vee ((p \land q) \lor r) \).

   • Consider when \( p \lor q \) is false: This happens only when \( p = \text{false} \) and \( q = \text{false} \), in which case the right expression \( ((p \land q) \lor r) \) can be true if \( r = \text{true} \).
   • Thus, with this choice, the expression becomes a tautology.
2. Checking other possibilities:
   • Choosing \( ▽ = \land \) with either \( Δ = \lor \) or \( Δ = \land \) doesn't lead to a tautology since not all combinations satisfy the condition.

Hence, the choice of operators that make the expression a tautology is \( ▽ = \lor \) and \( Δ = \land \). Under this choice, the expression \( (p \lor q) Δ r \equiv (p \land r) \lor q \), which represents \((pΔr)∨q\).

Thus, the logical equivalent expression is: (pΔr)∨q .