Determine each of the following.(a) {x ∈ Z | 0 < x, x2 ≤ 100}(b) |{x ∈ Z | 0 < x, x2 ≤ 100}|(c) |{x ∈ Z | x > 10, x2 ≤ 100}(d) {x ∈ Z | x > 10, x2 ≤ 100}|(e) | P(A) | , where A is the set from Part a.

Answer:(a) [tex]\{1,2,3,4,5,6,7,8,9,10\}[/tex](b) 10(c) [tex]\{\}[/tex](d) 0(e) 1024Step-by-step explanation:(a)A = {x ∈ Z | 0 < x, x² ≤ 100}We need to find all the elements of given set.The given conditions are[tex]0<x[/tex]                ... (1)[tex]x^2\leq 100[/tex]Taking square root on both sides.[tex]-\sqrt{100}\leq x\leq \sqrt{100}[/tex][tex]-10\leq x\leq 10[/tex]             .... (2)Using (1) and (2) we get[tex]0<x\leq 10[/tex]Since x ∈ Z,[tex]A=\{1,2,3,4,5,6,7,8,9,10\}[/tex](b)We need to find the value of  | {x ∈ Z | 0 < x, x² ≤ 100}| or |A|. It means have to find the number of elements in set A.[tex]|A|=10[/tex]| {x ∈ Z | 0 < x, x² ≤ 100}| = 10(c)B = {x ∈ Z | x > 10, x² ≤ 100}We need to find all the elements of given set.The given conditions are[tex]x>10[/tex]                ... (3)[tex]x^2\leq 100[/tex]It means [tex]-10\leq x\leq 10[/tex]             .... (4)Inequality (3) and (4) have no common solution, so B is null set or empty set.[tex]B=\{\}[/tex](d)We need to find the value of |{x ∈ Z | x > 10, x² ≤ 100}| or |B|. It means have to find the number of elements in set B.[tex]|B|=0[/tex]|{x ∈ Z | x > 10, x² ≤ 100}| = 0(e)We need to find the value of | P(A) |. P(A) is the power set of set A.Number of elements of a power set is[tex]N=2^n[/tex]where, n is the number of elements of set A.We know that the number of elements of set is 10. So the value of |P(A)| is[tex]|P(A)|=2^{10}[/tex][tex]|P(A)|=1024[/tex]Therefore |P(A)|=1024.