Which of the following predicate logic formulae/formula is/are CORRECT representation(s) of the statement: "Everyone has exactly one mother"?

The meanings of the predicates used are:

$\bullet$ mother $(y, x): y$ is the mother of $x$

$\bullet$ noteq $(x, y): x$ and $y$ are not equal

$A=\{0,1,2,3, \ldots\}$ is the set of non-negative integers. Let $F$ be the set of functions from $A$ to itself. For any two functions, $f_1, f_2 \in \mathrm{~F}$ we define

$$\left(f_1 \odot f_2\right)(n)=f_1(n)+f_2(n)$$

for every number $n$ in $A$. Which of the following is/are CORRECT about the mathematical structure $(\mathrm{F}, \odot)$ ?

Suppose a 5-bit message is transmitted from a source to a destination through a noisy channel. The probability that a bit of the message gets flipped during transmission is 0.01. Flipping of each bit is independent of one another. The probability that the message is delivered error-free to the destination is __________ ( (Rounded off to three decimal places)

Consider a probability distribution given by the density function $P(x)$.

$$P(x)=\left\{\begin{array}{cc} C x^2, & \text { for } 1 \leq x \leq 4 \\ 0, & \text { for } x<1 \text { or } x>4 \end{array}\right.$$

The probability that $x$ lies between 2 and 3, i.e., $P(2 \leq x \leq 3)$ is _________ (Rounded off to three decimal places)