G Tewani Solutions for Chapter: Mathematical Reasoning, Exercise 2: Archives

Statement $1: ~(p\leftrightarrow ~q)$ is equivalent to $p\leftrightarrow q$.

Statement $2: ~(p\leftrightarrow ~q)$ is a tautology.

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

The Boolean Expression $\left(p\wedge ~q\right)\vee q\vee \left(~p\wedge q\right)$ is equivalent to

The following statement $(p\to q)\to \left[(~p\to q)\to q\right]$ is

The boolean expression $~\left(p\vee q\right)\vee \left(~p\wedge q\right)$ is equivalent to