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Joint Entrance Examination
Mathematics
Mathematical Reasoning
Question 101
Question 101 of 103
Medium
MCQ
Statement \[\mathrm{(P \Rightarrow Q) \wedge(R \Rightarrow Q)}\] is logically equivalent to :
A
\[(P \Rightarrow R) \wedge(Q \Rightarrow R)\]
B
\[(P \Rightarrow R) \vee(Q \Rightarrow R)\]
C
\[(P \wedge R) \Rightarrow Q\]
D
\[(P \vee R) \Rightarrow Q\]
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Official solution
We have,
(
P
⇒
Q
)
∧
(
R
⇒
Q
)
(P \Rightarrow Q) \wedge(R \Rightarrow Q)
(
P
⇒
Q
)
∧
(
R
⇒
Q
)
\[
≡
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∼
P
∨
Q
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∧
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∼
R
∨
Q
)
≡
(
∼
P
∧
∼
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∨
Q
(by distributive law)
≡
∼
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∨
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≡
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⇒
Q
\begin{aligned} & \equiv(\sim P \vee Q) \wedge(\sim R \vee Q) \\\\ & \equiv(\sim P \wedge \sim R) \vee Q \text { (by distributive law) }\\\\ & \equiv \sim(P \vee R) \vee Q \\\\ & \equiv(P \vee R) \Rightarrow Q \end{aligned}
≡
(
∼
P
∨
Q
)
∧
(
∼
R
∨
Q
)
≡
(
∼
P
∧
∼
R
)
∨
Q
(by distributive law)
≡∼
(
P
∨
R
)
∨
Q
≡
(
P
∨
R
)
⇒
Q
\]
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