Mathematical logic-Past ( previous ) years question for entrance exam.

 Introduction:

    Hii,in this blog I write about past years question's of mathematical logic.This questions is very helpful for entrance exam like Mht cet,jee,Neet,etc.l relise that students during study find past years questions for practice.

* Important tips in logic

1)sentences which are imperative, exclamatory and interrogative are not statements in logic.

2)A statement pattern which is a contradiction is also called as fallacy.

3)If a statement is flase, its truth value is 'f'.

4)In logic 'but' & 'and' , 'while' is also conjunction.  

(I.e and=but=while)

*Important law in logic for entrance exam

1.Associative law

  (a) (p∨q)∨r=p∨(q∨r)=p∨q∨r

  (b) (p∧q)∧r=p∧(q∧r)=p∧q∧r

2.Distributive law

  (a) p∨(q∧r)=(p∨q)∧(p∨r)

  (b) p∧(q∨r)=(p∧q)∨(p∧r)

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3.Identity law 

  (a) p∨F=p

  (b) p∧F=F

  (c) p∨T= T

  (d) p∧T= p

4.Complement law

  (a) p∨~p=T

  (b) p∧~p=F

5.De Morgan's law 

  (a) ~(p∨q)=~p∧~q

  (b) ~(p∧q)=~p∨~q

6.Absorption law

  (a) p∨(p∧q)=p

  (b) p∧(p∨q)=p

7.Conditional law

  (a) p⇒q=~p∨q

  (b) p⇔q=(~p∨q)∧(~q∨p)

*Important Result

1. p⇔q=(~q⇒~p)

2. q⇒P=(~p⇒~q)

3. p⇔q=(p⇒q)∧(q⇒p)

* Past (previous) years questions in entrance exam:

(1)~(p∧q) is logically equivalent to                                                  (Mht cet 2007)

a)p∨q

b)p∧q

c)~p∨~q

d)~p∧~q

Ans:

@@@

   By De Morgan's law

 ~(p∧q)= ~p∨~q

  that's why (c)option is right.

(2) p∨(q∨r)=(p∨q)∨r.this law is called as                            [Mht cet 2006].

 a) Associative law

 b) Commutative law

 c) Distributive law

 d) Identity law

Ans:

  p∧(q∨r)=(p∨q)∨r  this is Associative law 

  that's why option (a) is right. 

(3) The equivalent form of the statement ~(p⇒~q) is.    [Mht cet 2019]

 a) p∧q

 b) ~p∨q

 c) p∧~q

 d) p∨~q

Ans:

  ~(p⇒~q)is equivalent to p∧q.

  by using 

~(p⇒~q)=~(~p∨~q)

             =~(~(p∧q))......(by negation)

             =p∧q

 that's why option (a) is right option.

(4) The statement pattern p∧(~p∨q)is       [Mht cet 2018]

 a) a tautology

 b) a contradiction

 c) equivalent to p∧q

 d) equivalent to p∨q

Ans:

       =p∧(~p∧q).....

       =(p∧~p)(p∧q)....

       =F∧(p∧q).(by identity law[p∧F=F])

       =F

 that's why option (b) is right.

(5) The statement pattern  (~p∧q) is logically equivalent to      [Mht cet 2017]

 a) (p∨q)∨~p

 b)(q∨p)∧~p

 c)(p∧q)⇒p

 d)(p∨q)⇒p

Ans:

   (p∨q)∧~p=(p∧~p)∨(q∧~p)

                   =F∨(~p∧q)

                   =~p∧q

 that's why option (b) is right.

(6) If 'c' denotes the  contradiction then the dual of the compound statement ~p∧(q∨c) is                     [Mht cet 2017]

 a) ~p∨(q∧t)

 b) ~p∧(q∨t)

 c)  p∨(~q∨t)

 d) ~p∧(q∧c) 

Ans:

  in dual of the statement the following changes occur

            (1) ∧ to ∨

            (2) ∨ to ∧

            (3) T to F

            (4) F to T

then, 

   dual of ~p∧(q∨c)

                =~p∨(q∧t)

 that's why option (a) is right.

(7) Let p and q are the true statements in logic which of the following statement pattern is a true? [Mht cet 2007]

 a) (p∧~q)⇒~q

 b) p∧~q

 c) (p∨q)⇒~p

 d) (~p∧~q)∧q

Ans:

       =(p∧~q)⇒~q

       =(T∧~T)⇒~T

       =(T∧F)⇒F

       =F⇒F

       =T

 that's why option (a) is right.

(8) negation of the statement, 'A is rich but silly' is.                         [Mht cet 2006]

 a) A is not rich or not silly.

 b) A is poor or not clever.

 c) A is rich but not silly.

 d) A is neither poor nor silly.

Ans:

  Let p: A is rich 

        q: A is silly

        ~(p∧q)=~p∨~q. .... (by negation)

that's why option (a) is right.

* Past (previous) years questions for practice:-

(1) Let r: he is  rich

              s: he is successful

              t: he is talented

Then the compound statement 'He is neither rich nor talented and hence he is not successful' is [Mht cet 2008]

 a) (~r∨~t)⇔~s

 b) (~r∧~t)⇒~s

 c) (~r∨~t)∧~s

 d) (~r∨t)∧~s

 

(2) Let p: RAM is lucky

              q: Ram works hard

              r: Ram Gates job

Then the compound statement 'If Ram is lucky and works hard, then he will get a job' is [Mht cet 2006]

 a) (p∧q)⇒r

 b) ~(p∧q)⇒r

 c) (p∨q)⇒r

 d) ~(p∨q)⇒r

(3) If p: every square is a rectangle

           q: every rambus is it then truth value of p⇒q and p⇔q are - - - and - - - respectively.

 a) F,F

 b) T,F

 c) F,T

 d) T,T

(4) The statement pattern (p∧q)∧[~r∨(p∧q)]∨(~p∧q) is equivalent to.                     [Mht cet 2019]

 a) r

 b) p

 c) q

 d) p∧q

 

(5) The negation of the statement, "Getting above 95% Mark's is necessary condition for Hema to get the admission in good college".[Mht cet 2018]

 a) Hema gets above 95% marks but           she does not get the admission in good college.

 b) Hema does not get above 95% marks and she gets admission in good college.

 c) If Hema does not get the above 95% then she will not get admission in good. college.

 d) Hema does not get above 95% marks or she gets the admission in good college.

(6) Which of the following statement pattern is a Tautology? [Mht cet 2017]

 a) p∨(q⇒p)

 b) ~q⇒~p

 c) (q⇒p)∨(~p⇔q)

 d) p∧~p

 Answer's of past years questions of practice:

(1)-b

(2)-a

(3)-d

(4)-a

(5)-b

(6)-c

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