Last Updated : 04 Jan, 2019
How many given logical equivalence(s) is/are false? 
(I) {\\displaystyle \\neg (p\\iff q)\\equiv p\\iff \\neg q}
(II) {\\displaystyle (p\\implies q)\\wedge (p\\implies r)\\equiv p\\implies (q\\vee r)} 
(III) {\\displaystyle (p\\implies q)\\vee (p\\implies r)\\equiv p\\implies (q\\wedge r)}  
(IV)  {\\displaystyle (p\\implies r)\\wedge (q\\implies r)\\equiv (p\\vee q)\\implies r} 
(V) {\\displaystyle (p\\implies r)\\vee (q\\implies r)\\equiv (p\\vee q)\\implies r}
(A) 0 (B) 2 (C) 3 (D) None of these

Answer: (C)

Explanation: Correct logical equivalences are as : (I) {\\displaystyle \\neg (p\\iff q)\\equiv p\\iff \\neg q} (II) {\\displaystyle (p\\implies q)\\wedge (p\\implies r)\\equiv p\\implies (q\\wedge r)} (III) {\\displaystyle (p\\implies q)\\vee (p\\implies r)\\equiv p\\implies (q\\vee r)} (IV) {\\displaystyle (p\\implies r)\\wedge (q\\implies r)\\equiv (p\\vee q)\\implies r} (V) {\\displaystyle (p\\implies r)\\vee (q\\implies r)\\equiv (p\\wedge q)\\implies r} Therefore, only equivalences (II), (III), and (V) are false. So, answer is 3.

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