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Which of the following pairs of propositions are not logically equivalent?

  1. $((p \rightarrow r) \wedge (q \rightarrow r))$ and $((p \vee q) \rightarrow r)$
  2. $p \leftrightarrow q$ and $(\neg p \leftrightarrow \neg q)$
  3. $((p \wedge q) \vee (\neg p \wedge \neg q))$ and $p \leftrightarrow q$
  4. $((p \wedge q) \rightarrow r)$ and $((p \rightarrow r) \wedge (q \rightarrow r))$
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Logical Equivalences Involving Biconditional Statements.


p ↔ q ≡ (p → q) ∧ (q → p)
p ↔ q ≡ ¬p ↔¬q                        ( option 2)
p ↔ q ≡ (p ∧ q) ∨ (¬p ∧¬q)       (option 3)

 

Logical Equivalences Involving Conditional Statements

p → q ≡ ¬p ∨ q
p → q ≡ ¬q →¬p
¬(p → q) ≡ p ∧¬q
(p → q) ∧ (p → r) ≡ p → (q ∧ r)
(p → r) ∧ (q → r) ≡ (p ∨ q) → r  ( option 1)
(p → q) ∨ (p → r) ≡ p → (q ∨ r)
(p → r) ∨ (q → r) ≡ (p ∧ q) → r (in option 4  there is and instead of or )  hence correct ans is 4

(we can always  check via truth tables or by removing --> and <-> if above formulas are not learned properly)

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