Let $$X$$ be a Gaussian random variable with mean $$0$$ and variance $${\sigma ^2}$$ . Let $$Y=max(X,0)$$ where $$max(a, b)$$ is the maximum of $$a$$ and $$b$$. The median of $$Y$$ is ___________.

Let $$A$$ be $$n\,\, \times \,\,n$$ real valued square symmetric matrix of rank $$2$$ with $$\sum\limits_{i = 1}^n {\sum\limits_{j = 1}^n {A_{ij}^2 = 50.} } $$
Consider the following statements.
$$(I)$$ One eigenvalue must be in $$\left[ { - 5,5} \right]$$
$$(II)$$ The eigenvalue with the largest magnitude must be strictly greater than $$5$$
Which of the above statements about engenvalues of $$A$$ is/are necessarily correct?

The statement $(\neg p) \Rightarrow(\neg q)$ is logically equivalent to which of the statements below?

I. $\quad p \Rightarrow q$

II. $q \Rightarrow p$

III. $(\neg q) \vee p$

IV. $(\neg p) \vee q$

Consider the first-order logic sentence $F: \forall x(\exists y R(x, y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$?

I. $\quad \exists y(\exists x R(x, y))$

II. $\quad \exists y(\forall x R(x, y))$

III. $\forall y(\exists x R(x, y))$

IV. $\neg \exists x(\forall y \neg R(x, y))$