01. Let 𝑈 = {−2; 1; 0;1/2;3/4; 2;10/3; 4; −5}. Explain the elements of each of the following sets: a) {𝑥 ∈ 𝑈/𝑥 − 4 < 0} b) {𝑥 ∈ 𝑈/𝑥 2 − 6𝑥 + 8 < 0} c) {𝑥 ∈ 𝑈/4 − 𝑥 ≤ 0} d) {𝑥 ∈ 𝑈/𝑥 2 + 1 ≤ 0}

Given set 𝑈 = {−2; 1; 0; 1/2; 3/4; 2; 10/3; 4; −5}

a) {𝑥 ∈ 𝑈/𝑥 - 4 < 0}
To solve this, first, we need to find the elements in 𝑈 where 𝑥 - 4 < 0.
𝑥 - 4 < 0
𝑥 < 4

The elements in set 𝑈 that satisfy 𝑥 < 4 are {-2; 1; 0; 1/2; 3/4; 2, 10/3}.

b) {𝑥 ∈ 𝑈/𝑥^2 - 6𝑥 + 8 < 0}
To solve this, first, we need to find the elements in 𝑈 where 𝑥^2 - 6𝑥 + 8 < 0.
(𝑥 - 4)(𝑥 - 2) < 0
The solutions to this inequality are 𝑥 ∈ (2, 4).
Thus, the elements in set 𝑈 that satisfy this inequality are {3/4}.

c) {𝑥 ∈ 𝑈/4 - 𝑥 ≤ 0}
To solve this, first, we need to find the elements in 𝑈 where 4 - 𝑥 ≤ 0.
4 - 𝑥 ≤ 0
𝑥 ≥ 4
The only element in set 𝑈 that satisfies this inequality is {4}.

d) {𝑥 ∈ 𝑈/𝑥^2 + 1 ≤ 0}
To solve this, first, we need to find the elements in 𝑈 where 𝑥^2 + 1 ≤ 0.
This inequality has no real solutions because 𝑥^2 + 1 is always greater than 0 for all real values of 𝑥.
Therefore, there are no elements in 𝑈 that satisfy this inequality.

\textbf{Answer:}
a) {-2; 1; 0; 1/2; 3/4; 2; 10/3}
b) {3/4}
c) {4}
d) No elements in set 𝑈 satisfy the inequality.