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Question

Which of the following statements is a tautology?


A

(~qp)q

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B

(~qp)(p~p)

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C

(~qp)(p~p)

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D

pq~pq

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Solution

The correct option is C

(~qp)(p~p)


Explanation for correct option:

A tautology is a compound statement that is always true.

Signs of tautology:

Symbol Meaning
~Not
And
Or
Implies or If-then

Make truth table to check the relation (~qp)(p~p)

p~pq~q(~qp)(p~p)(~qp)(p~p)
TFTFFTT
TFFTTTT
FTTFFTT
FTFTFTT

Since the results are true. (~qp)(p~p) is tautology.

Explanation for incorrect options:

Option (A):

p~pq~q(~qp)(~qp)q
TFTFFF
TFFTTT
FTTFFF
FTFTFF

Since all results are not true. So (~qp)q is not a tautology.

Option (B):

p~pq~q(~qp)(p~p)(~qp)(p~p)
TFTFFTF
TFFTTTT
FTTFFTF
FTFTFTF

Since all results are not true. So (~qp)(p~p) is not a tautology.

Option (D):

p~pq~qpq~pqpq~pq
TFTFTFF
TFFTFTF
FTTFFTF
FTFTFTF

Since all results are false. So, it is a contradiction.

Thus it is not a tautology

Hence option (C) is correct.


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