Mathematics is interesting is not a logical sentence. It may be interesting for some persons are may not be interesting for others.
$$\therefore $$ This is not a propositions.
13.
The negation of the statement $$\left( {p \wedge q} \right) \to \left( { \sim p \vee r} \right){\text{is}}$$
When $$p$$ is false and $$q$$ is true, then $$p \wedge q$$ is false, $$p \vee \sim q$$ is false. ($$\because $$ both $$p$$ and $$ \sim q$$ are false)
and $$q \Rightarrow p$$ is also false,
only $$p \Rightarrow q$$ is true.
17.
Consider the following statements
$$p :$$ A tumbler is half empty.
$$q :$$ A tumbler is half full.
Then, the combination form of “$$p$$ if and only if $$q\,$$” is
A
a tumbler is half empty and half full
B
a tumbler is half empty if and only if it is half full
C
Both $$\left( A \right){\text{ and }}\left( B \right)$$
D
None of the above
Answer :
a tumbler is half empty if and only if it is half full
The given statements are
$$p :\,$$ A tumbler is half empty.
$$q :\,$$ A tumbler is half full.
We know that, if the first statement happens, then the second happens and also if the second happens, then the first happens. We can express this fact as
If a tumbler is half empty, then it is half full.
If a tumbler is half full, then it is half empty.
We combine these two statements and get the following. A tumbler is half empty, if and only if it is half full.
18.
If $$p :$$ It is snowing, $$q :$$ I am cold, then the compound statement "It is snowing and it is not that I am cold" is given by
$${p \Rightarrow q}$$ is logically equivalent to $${ \sim p \Rightarrow \, \sim q}$$
$$\therefore \left( {p \Rightarrow q} \right) \Leftrightarrow \left( { \sim q \Rightarrow \, \sim p} \right)\,$$ is a tautology but not a contradiction.