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Joint Entrance Examination
Mathematics
Mathematical Reasoning
Question 81
Question 81 of 103
Medium
MCQ
The statement \[(p \Rightarrow q) \vee(p \Rightarrow r)\] is NOT equivalent to
A
\[(p \wedge(\sim r)) \Rightarrow q\]
B
\[(\sim q) \Rightarrow((\sim r) \vee p)\]
C
\[p \Rightarrow(q \vee r)\]
D
\[(p \wedge(\sim q)) \Rightarrow r\]
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Official solution
(
A
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(
p
∧
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∼
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⇒
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(\mathrm{A})(p \wedge(\sim r)) \Rightarrow q
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⇒
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\[
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\begin{aligned} &\sim(p \wedge \sim r) \vee q \\\\ &\equiv(\sim p \vee r) \vee q \\\\ &\equiv \sim p \vee(r \vee q) \\\\ &\equiv p \rightarrow(q \vee r) \\\\ &\equiv(p \Rightarrow q) \vee(p \Rightarrow r) \end{aligned}
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\] (C)
p
⇒
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p \Rightarrow(q \vee r)
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\[
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\begin{aligned} &\equiv \sim p \vee(q \vee r) \\\\ &\equiv(\sim p \vee q) \vee(\sim p \vee r) \\\\ &\equiv(p \rightarrow q) \vee(p \rightarrow r) \end{aligned}
≡∼
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\] (D)
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(p \wedge \sim q) \Rightarrow r
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\[ \equiv p \Rightarrow(q \vee r) \] \[
≡
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∨
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\begin{aligned} & \equiv(p \Rightarrow q) \vee(p \Rightarrow r) \end{aligned}
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\]
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