Which of the following pairs of propositions are not logically equivalent?
Which of the following pairs of propositions are not logically equivalent?
Answer: D. p→r∧q→r and p∨q→r — (d) The first proposition, p→r ∧ q→r, is interpreted as (p→r) ∧ (q→r) due to operator precedence (∧ has higher precedence than →). Simplifying (p→r) ∧ (q→r):…
- A.
p∧q→r and ~r→(~p∨~q)
- B.
p↔q and ~p↔~q
- C.
(p→q)⋀(q→p) and p↔q
- D.
p→r∧q→r and p∨q→r
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Correct answer: D
(d) The first proposition, p→r ∧ q→r, is interpreted as (p→r) ∧ (q→r) due to operator precedence (∧ has higher precedence than →).
Simplifying (p→r) ∧ (q→r):
=> (¬p∨r) ∧ (¬q∨r)
=> (¬p∧¬q)∨r
The second proposition, p∨q→r, simplifies to:
=> ¬(p∨q)∨r
=> (¬p∧¬q)∨r
Both expressions simplify to (¬p∧¬q)∨r, so they are logically equivalent.
Similarly, we can prove that the rest of the propositions are equivalent.
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