Q: The Boolean expression (p q) ( p q) is equivalent to

aligned & Given, (p q) ( p q) \\ & ( p q) ( p q) (By De Morgan's Law) \\ & p ( q q) (Distributive Law) \\ & p t (By complement law, where t= tautology) \\ & p (Identity law) aligned

Q: Consider the following statements : A: Rishi is a judge. B : Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" i

aligned & 2(1+ t) t 2 ^2 t = 3 \\ & t= 6 , y_0=27 aligned

Question 1

The Boolean expression (p q) ( p q) is equivalent to

Accepted Answer

aligned & Given, (p q) ( p q) \ & ( p q) ( p q) (By De Morgan's Law) \ & p ( q q) (Distributive Law) \ & p t (By complement law, where t= tautology) \ & p (Identity law) aligned

Question 2

If p: 2 is an even number q : 2 is a prime number, r : 2+2=2^2 Then the symbolic statement p (q r) means

Accepted Answer

p ( q r ). If 2 is even number then 2 is a prime number or 2+2=2^2.

Question 3

Consider the following statements : A: Rishi is a judge. B : Rishi is honest. C : Rishi is not arrogant. The negation of the statement "if Rishi is a judge and he is not arrogant, then he is honest" i

Accepted Answer

aligned & 2(1+ t) t 2 ^2 t = 3 \ & t= 6 , y_0=27 aligned

Question 4

If p : It raining today, q : I go to school, r : I shall meet any friends and s : I shall go for a movie, then which of the following is the proposition : 'If it does not rain or if I do not go to sch

Accepted Answer

Correct result is ( p q ) ( r s ) So, (p q) (r s)

Question 5

Let p, q, r be three logical statements. Consider the compound statements S_1:(( p) q) (( p) r) and S_2: p (q r) Then, which of the following is NOT true?

Accepted Answer

Given statement S_1:( p q) ( p r) p (q r) S_2: p (q r) p (q r) By conditional law S_1 S_2

Question 6

Consider the following two propositions: P _1: (p q) P_2:(p q) (( p) q) If the proposition p (( p) q) is evaluated as FALSE, then :

Accepted Answer

Given propositions P _1: ( p q ) P _2:(p q) (( p) q) Required table is shown below. $ array |l|l|l|l|l|l|l|l|l|l|l| p & q & p & q & p q & p ( (p q) & p q & (p q) & p q & p_2 \ T & T & F & F & T & T & F & T & F & F \ T & F & F & T & F & F & T & F & T & F \ F & T & T & F & T & T & T & F & F & F \ F & F & T & T & T & T & T & F & F & F \ array $ $p ( p q)$ is $F$ when $p$ is true $q$ is false From table $P_1 ~\&~ P_2$ both are false

Question 7

The negation of the Boolean expression (( q) p) (( p) q) is logically equivalent to

Accepted Answer

Given boolen expression are aligned & p q p q and q p (p q) \ & Negation of (p q) (p q) \ & is (p q) ( (p q)) i.e., (p q) aligned

Question 8

Negation of the Boolean expression p ( q p ) is

Accepted Answer

Given expression is p (q p) ( p ( q p )) ( p q )=( p q ) ( q p ) ( p ( q p )) =( p ( q p )) (( q p ) p ) ( p ( q p ))= p ( q p ) =(p p) q=c (q p) p=( q p) p = p ( q p) =( p q ) ( p p )= p q (p ( q p))=c ( p q)= p q

Question 9

The statement p: For any real numbers x, y if x=y, then 2 x+a=2 y+a when a Z .

Accepted Answer

We prove the statement p is true by contrapositive method and by direct method. Direct method: For any real number x and y, array l x=y \ 2 x=2 y \ 2 x+a=2 y+a a for some a Z array Contrapositive method: The contrapositive statement of p is "For any real numbers x, y if 2 x + a 2 y + a , where a Z , then x y ." Given, 2 x+a 2 y+a array l 2 x 2 y \ x y array Hence, the given statement is true.

Mathematical Reasoning — Previous Year Questions & Solutions