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Question
Mathematics
Which of the following statements is a tautology?
Q. Which of the following statements is a tautology?
156
113
JEE Main
JEE Main 2023
Mathematical Reasoning
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A
p
∨
(
p
∧
q
)
B
(
p
∧
(
p
→
q
))
→∼
q
C
(
p
∧
q
)
→
(
∼
(
p
)
→
q
)
D
p
→
(
p
∧
(
p
→
q
))
Solution:
(i)
p
→
(
p
Λ
(
p
→
q
))
(
∼
p
)
V
(
p
Λ
(
∼
p
V
q
))
(
∼
p
)
V
(
f
V
(
p
Λ
q
))
∼
p
V
(
p
Λ
q
)
=
(
∼
p
V
p
)
Λ
(
∼
p
V
q
)
=∼
p
V
q
(ii)
(
p
Λ
q
)
→
(
∼
p
→
q
)
∼
(
p
Λ
q
)
V
(
p
V
q
)
=
t
{
a
,
b
,
d
}
V
{
a
,
b
,
c
}
=
V
Tautology
(iii)
(
p
Λ
(
p
→
q
))
→∼
q
∼
(
p
Λ
(
∼
p
V
q
))
V
∼
q
=∼
(
p
Λ
q
)
V
∼
q
=∼
p
V
∼
q
Not tantology
(iv)
p
V
(
p
Λ
q
)
=
p
Not tautology.