We are asked to list the elements of the following sets: (a) {$x: x$ is an integer greater than $-2$ but less than $9$}. (b) {$x: x$ is an integer} $\cap$ {$0, \sqrt{3}, \pi, 2i$}. (c) {$x^2 + 1: x \in A$} where $A = \{-2, -1, 0, 1, 2\}$. (d) {$\sqrt{x + 2}: x \in B$} where $B = \{-3, -4, 0, 1, 2\}$.

1. Problem Description

We are asked to list the elements of the following sets:

(a) {$x: x$ is an integer greater than $-2$ but less than $9$}.

(b) {$x: x$ is an integer} $\cap$ {$0, \sqrt{3}, \pi, 2i$}.

(c) {$x^2 + 1: x \in A$} where $A = \{-2, -1, 0, 1, 2\}$.

(d) {$\sqrt{x + 2}: x \in B$} where $B = \{-3, -4, 0, 1, 2\}$.

2. Solution Steps

(a) The integers greater than $-2$ but less than $9$ are $-1, 0, 1, 2, 3, 4, 5, 6, 7, 8$.

(b) The set {$x: x$ is an integer} is the set of all integers. We are looking for the intersection of this set with the set {$0, \sqrt{3}, \pi, 2i$}.

The only integer in the set {$0, \sqrt{3}, \pi, 2i$} is $0$. $\sqrt{3}$ and $\pi$ are irrational numbers and $2i$ is an imaginary number. Therefore, the intersection is {$0$}.

(c) We have $A = \{-2, -1, 0, 1, 2\}$. We need to find the set {$x^2 + 1: x \in A$}. We compute $x^2 + 1$ for each $x$ in $A$.

If $x = -2$, $x^2 + 1 = (-2)^2 + 1 = 4 + 1 = 5$.

If $x = -1$, $x^2 + 1 = (-1)^2 + 1 = 1 + 1 = 2$.

If $x = 0$, $x^2 + 1 = (0)^2 + 1 = 0 + 1 = 1$.

If $x = 1$, $x^2 + 1 = (1)^2 + 1 = 1 + 1 = 2$.

If $x = 2$, $x^2 + 1 = (2)^2 + 1 = 4 + 1 = 5$.

Therefore, the set is {$5, 2, 1, 2, 5$} which is equal to {$1, 2, 5$}.

(d) We have $B = \{-3, -4, 0, 1, 2\}$. We need to find the set {$\sqrt{x + 2}: x \in B$}. We compute $\sqrt{x + 2}$ for each $x$ in $B$.

If $x = -3$, $\sqrt{x + 2} = \sqrt{-3 + 2} = \sqrt{-1} = i$.

If $x = -4$, $\sqrt{x + 2} = \sqrt{-4 + 2} = \sqrt{-2} = i\sqrt{2}$.

If $x = 0$, $\sqrt{x + 2} = \sqrt{0 + 2} = \sqrt{2}$.

If $x = 1$, $\sqrt{x + 2} = \sqrt{1 + 2} = \sqrt{3}$.

If $x = 2$, $\sqrt{x + 2} = \sqrt{2 + 2} = \sqrt{4} = 2$.

Therefore, the set is {$i, i\sqrt{2}, \sqrt{2}, \sqrt{3}, 2$}.

3. Final Answer

(a) {$-1, 0, 1, 2, 3, 4, 5, 6, 7, 8$}

(b) {$0$}

(c) {$1, 2, 5$}

(d) {$i, i\sqrt{2}, \sqrt{2}, \sqrt{3}, 2$}