Let $P(x)$ be an arbitrary predicate over the domain of natural numbers. Which ONE of the following statements is TRUE?

Let $F$ be the set of all functions from $\{1, \ldots, n\}$ to $\{0,1\}$. Define the binary relation $\preccurlyeq$ on $F$ as follows:

$\forall f . g \in F, f \preccurlyeq g$ if and only if $\forall x \in\{1, \ldots, n\}, f(x) \leq g(x)$, where $0=1$.

Which of the following statement(s) is/are TRUE? re TRUE?

Consider a system of linear equations $P X=Q$ where $P \in \mathbb{R}^{3 \times 3}$ and $Q \in \mathbb{R}^{3 \times 3}$. Suppose $P$ has an $L U$ decomposition, $P=L U$, where

$$L=\left[\begin{array}{ccc} 1 & 0 & 0 \\ l_{21} & 1 & 0 \\ l_{31} & l_{32} & 1 \end{array}\right] \text { and } u=\left[\begin{array}{ccc} u_{11} & u_{12} & u_{13} \\ 0 & u_{22} & u_{23} \\ 0 & 0 & u_{33} \end{array}\right]$$

Which of the following statement(s) is/are TRUE?

A quadratic polynomial $(x-\alpha)(x-\beta)$ over complex numbers is said to be square invariant if $(x-\alpha)(x-\beta)=\left(x-\alpha^2\right)\left(x-\beta^2\right)$. Suppose from the set of all square invariant quadratic polynomials we choose one at random.

The probability that the roots of the chosen polynomial are equal is (rounded off to one decimal place)