Question

Determine each of the following.

Answer 1

Answer:

(a) $\{1,2,3,4,5,6,7,8,9,10\}$

(b) 10

(c) $\{\}$

(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

$0$                ... (1)

$x^2\leq 100$

Taking square root on both sides.

$-\sqrt{100}\leq x\leq \sqrt{100}$

$-10\leq x\leq 10$             .... (2)

Using (1) and (2) we get

$0$

Since x ∈ Z,

$A=\{1,2,3,4,5,6,7,8,9,10\}$

(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.

$|A|=10$

| {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

$x>10$                ... (3)

$x^2\leq 100$

It means

$-10\leq x\leq 10$             .... (4)

Inequality (3) and (4) have no common solution, so B is null set or empty set.

$B=\{\}$

(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.

$|B|=0$

|{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

$N=2^n$

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

$|P(A)|=2^{10}$

$|P(A)|=1024$

Therefore |P(A)|=1024.