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Joint Entrance Examination
Mathematics
Mathematical Reasoning
Question 4
Question 4 of 103
Medium
MCQ
Which of the following statements is a tautology?
A
p
→
(
p
∧
(
p
→
q
)
)
p\to(p\land(p\to q))
p
→
(
p
∧
(
p
→
q
))
B
(
p
∧
q
)
→
(
¬
p
→
q
)
(p\land q)\to(\lnot p\to q)
(
p
∧
q
)
→
(
¬
p
→
q
)
C
(
p
∧
(
p
→
q
)
)
→
¬
q
(p\land(p\to q))\to\lnot q
(
p
∧
(
p
→
q
))
→
¬
q
D
p
∨
(
p
∧
q
)
p\lor(p\land q)
p
∨
(
p
∧
q
)
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Official solution
∼
p
→
q
≡
∼
(
∼
p
)
∨
q
≡
p
∨
q
p
∧
q
→
(
∼
p
→
q
)
≡
p
∧
q
→
(
p
∨
q
)
≡
∼
(
p
∧
q
)
∨
(
p
∨
q
)
≡
(
∼
p
∨
∼
q
)
∨
(
p
∨
q
)
≡
(
∼
p
∨
(
p
∨
q
)
)
∨
(
∼
q
∨
(
p
∨
q
)
)
≡
T
∨
T
≡
T
\begin{aligned} & \sim p \rightarrow q \equiv \sim(\sim p) \vee q \equiv p \vee q \\\\ & p \wedge q \rightarrow(\sim p \rightarrow q) \\\\ & \equiv p \wedge q \rightarrow(p \vee q) \\\\ & \equiv \sim(p \wedge q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee \sim q) \vee(p \vee q) \\\\ & \equiv(\sim p \vee(p \vee q)) \vee(\sim q \vee(p \vee q)) \\\\ & \equiv T \vee T \\\\ & \equiv T\end{aligned}
∼
p
→
q
≡∼
(
∼
p
)
∨
q
≡
p
∨
q
p
∧
q
→
(
∼
p
→
q
)
≡
p
∧
q
→
(
p
∨
q
)
≡∼
(
p
∧
q
)
∨
(
p
∨
q
)
≡
(
∼
p
∨
∼
q
)
∨
(
p
∨
q
)
≡
(
∼
p
∨
(
p
∨
q
))
∨
(
∼
q
∨
(
p
∨
q
))
≡
T
∨
T
≡
T
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