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Question

If p,q,r are any three logical statements, then which one of the following is correct?


A

~[p(~q)](~p)q

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B

~(pq)(~r)(~p)(~q)(~r)

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C

~[p(~q)](~p)q

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D

~[p(~q)](~p)~q

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Solution

The correct option is C

~[p(~q)](~p)q


Explanation for the correct option

Check for option (C) which is ~p~q~pq

First, solve for LHS and check if it's equal to RHS.

~p~q=~p~~q(usingDemorgansLaw)=~pq(usingDoubleNegationLaw)

From this one can conclude that,

L.H.S=R.H.S

Therefore option (C) is correct.

Explanation for incorrect options

Check for option (A) which is~[p(~q)](~p)q

Solve for LHS and check if it's equal to RHS.

~p~q=~p~~q(usingDemorgansLaw)=~pq(usingDoubleNegetarionLaw)

Here, L.H.SR.H.S

Therefore, option (A) is incorrect.

Check for option (B) which is~(pq)(~r)(~p)(~q)(~r)

Solve for LHS and check if it's equal to RHS.

~(pq)(~r)=(~p)(~q)(~r)(usingDemorgansLaw)=(~p)(~q)(~r)(usingDoubleNegetarionLaw)

Here as well, L.H.SR.H.S

Therefore, option (B) is incorrect.

Check for option (D) which is~[p(~q)](~p)~q

Solve for LHS and check if it's equal to RHS.

~[p(~q)]=(~p)~(~q)(usingDemorgansLaw)=(~p)q(usingDoubleNegetarionLaw)

We can see that L.H.SR.H.S

Therefore, option (D) is incorrect.

Hence option (C) is the correct option.


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