Respuesta :
Answer:
(a) [tex]\{1,2,3,4,5,6,7,8,9,10\}[/tex]
(b) 10
(c) [tex]\{\}[/tex]
(d) 0
(e) 1024
Step-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.
