Mathematical Reasoning

Consider the following three statements :
P : 5 is a prime number
Q : 7 is a factor of 192
R : L.C.M. of 5 and 7 is 35
Then the truth value of which one of the following statements is true ?

Which one of the following Boolean expressions is a tautology?

The negation of the Boolean expression ~ s \(\vee\) (~r \(\wedge\) s) is equivalent to :

If p \(\to\) (\(\sim\) p\(\vee\) \(\sim\) q) is false, then the truth values of p and q are respectively :

If (p \(\wedge\) \(\sim\) q) \(\wedge\) (p \(\wedge\) r) \(\to\) \(\sim\) p \(\vee\) q is false, then the truth values of \(p, q\) and \(r\) are, respectively :

Consider the following two statements :
Statement p :
The value of sin 120o can be derived by taking \(\theta = {240^o}\) in the equation
2sin\({\theta \over 2} = \sqrt {1 + \sin \theta } - \sqrt {1 - \sin \theta }\)
Statement q :
Th...

The Boolean expression
\(\sim \left( {p \vee q} \right) \vee \left( { \sim p \wedge q} \right)\) is equvalent to :

Contrapositive of the statement
‘If two numbers are not equal, then their squares are not equal’, is :

The proposition \(\left( { \sim p} \right) \vee \left( {p \wedge \sim q} \right)\) is equivalent to :

The following statement
\(\left( {p \to q} \right) \to \left[ {\left( { \sim p \to q} \right) \to q} \right]\) is :

Consider the following two statements :
P :     If 7 is an odd number, then 7 is
divisible by 2.
Q :    If 7 is a prime number, then 7 is an
odd number
If  V1 is the truth value of the contrapositive of P and V2 is the truth value of contr...

The contrapositive of the following statement,
“If the side of a square doubles, then its area increases four times”, is :

The Boolean expression
\(\left( {p \wedge \sim q} \right) \vee q \vee \left( { \sim p \wedge q} \right)\) is equivalent to :

The negation of \(\sim s \vee \left( { \sim r \wedge s} \right)\) is equivalent to :

The statement \(\sim \left( {p \leftrightarrow \sim q} \right)\) is :

Consider :
Statement − I : \(\left( {p \wedge \sim q} \right) \wedge \left( { \sim p \wedge q} \right)\) is a fallacy.
Statement − II :\(\left( {p \to q} \right) \leftrightarrow \left( { \sim q \to \sim p} \right)\) is a tautology.

The negation of the statement “If I become a teacher, then I will open a school” is :

Consider the following statements
P : Suman is brilliant
Q : Suman is rich
R : Suman is honest
The negation of the statement,
“Suman is brilliant and dishonest if and only if Suman is rich” can be
expressed as :

Let S be a non-empty subset of R. Consider the following statement:
P : There is a rational number x ∈ S such that x > 0.
Which of the following statements is the negation of the statement P?

Statement-1 : \(\sim \left( {p \leftrightarrow \sim q} \right)\) is equivalent to \({p \leftrightarrow q}\).
Statement-2 : \(\sim \left( {p \leftrightarrow \sim q} \right)\) is a tautology.

The statement \(p \to \left( {q \to p} \right)\) is equivalent to

Let p be the statement “x is an irrational number”, q be the statement “y is a transcendental number”,
and r be the statement “x is a rational number iff y is a transcendental number”.
Statement –1: r is equivalent to either q or p.
State...