Embibe Experts Solutions for Chapter: Mathematical Reasoning, Exercise 3: Exercise-3

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 an irrational number iff $y$ is a transcendental number".

Statement-$1:r$ is equivalent to either $q$ or $p$.

Statement-$2:r$ is equivalent to $~\left(p\iff ~q\right)$.

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

The only statement among the following that is a tautology, is

The negation of the statement

"If I become a teacher, then I will open a school", is;

Consider
Statement $\mathrm{I}$ $\left(p\wedge ~q\right)\wedge \left(~p\wedge q\right)$ is a fallacy.
Statement $\mathrm{II}$ $\left(p\to q\right)\leftrightarrow \left(~q\to ~p\right)$ is a tautology.

The statement $~\left(p\leftrightarrow ~q\right)$ is

The negation of $~s\vee \left(~r\wedge s\right)$ is equivalent to

Consider the following three statements:

$P:5$ is a prime number

$Q:7$ is a factor of $192$

$R: \mathrm{L}.\mathrm{C}.\mathrm{M}.$ of $5$ and $7$ is $35$.

Then the truth value of which one of the following statements is true?