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 if $$f y$$  is a transcendental number”.
Statement - 1 : $$r$$ is equivalent to either $$q$$ or $$p$$
Statement - 2 : $$r$$ is equivalent to $$ \sim \left( {p \leftrightarrow \sim q} \right).$$

Solution :
$$p$$ : $$x$$ is an irrational number
$$q$$ : $$y$$ is a transcendental number
$$r$$ : $$x$$ is a rational number if $$f y$$ is a transcendental number.
clearly $$r: \sim p \leftrightarrow q$$
Let us use truth table to check the equivalence of $$‘r’$$ and $$‘q$$ or $$p’,'r'$$  and $$ \sim \left( {p \leftrightarrow \sim q} \right)$$

				1	2		3
$$p$$	$$q$$	$$ \sim p$$	$$ \sim q$$	$$ \sim p \leftrightarrow q$$	$$q{\text{ or }}p$$	$$p \leftrightarrow \, \sim q$$	$$ \sim \left( {p \leftrightarrow \, \sim q} \right)$$
T	T	F	F	F	T	F	T
T	F	F	T	T	T	T	F
F	T	T	F	T	T	T	F
F	F	T	T	F	F	F	T

From columns (1), (2) and (3), we observe, none of the these statements are equivalent to each other.
∴ Statement las well as statement 2 both are false.
∴ None of the options is correct.