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## Here's the Solution to this Question

Solution.

A.

$\{x| x \text{ is real numbers and } x^2=1\}=\{-1,1\}.$

$\{x| x \text{ is an integer and } -4

B.

$\{a,e,i,o,u\}=\{x| x\text{ is vowels letters, without 'y' }\}.$

$\{-2,-1,0,1,2\}=\{x \in Z| |x|=2\}.$

C.

$A=(a,b,c), B=(x,y), C=(0,1).$

$A \text{U} C=\{a,b,c,0,1\}. \newline C \times B=\{(0,x),(0,y),(1,x),(1,y)\}. \newline B-A=\{x,y\}. \newline (A \cap C)\text{U} B=\{x,y\}.$

D.

$A_n=2\cdot 3^n+5. \newline A_0=7. \newline A_5=491. \newline A_3=59. \newline A_8=13127. \newline A_2=23. \newline \sum A_n=\infty.$

E.

$X=\{-1,0,1,2,3,4,5\}. \newline R=\{(x,y)|x\leq y\}.$

F.

$R=\{(-1,0),(-1,1),(-1,2),(-1,3),(-1,4),(-1,5),\newline (0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),\newline (2,5),(3,4),(3,5),(4,5),(5,5),(4,4),(3,3),(2,2),(1,1),(0,0),(-1,-1)\}.$

G.

Domain($R$ )=$\{-1,0,1,2,3,4,5\}.$

H.

Range($R$ )=$\{-1,0,1,2,3,4,5\}.$

I. J.

Properties of the Relation:

Reflexivity

Irreflexivity

Symmetry

Antysymmetry

Asymmetry

Transitivity.