Question

Determine each of the following.

(a) 0 < x, x2 ≤ 100

(b) | 0 < x, x2 ≤ 100|

(c) | x > 10, x2 ≤ 100

(d) x > 10, x2 ≤ 100|

(e) | P(A) | , where A is the set from Part a.

Answer 1

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

(b) 10

(c)
`\{\}`

(d) 0

(e) 1024

Explanation:

(a)

A = 0 < x, x² ≤ 100

We need to find all the elements of given set.

The given conditions are

`0<x` ... (1)

`x^2\leq 100`

Taking square root on both sides.

`-√(100)\leq x\leq √(100)`

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

Using (1) and (2) we get

`0<x\leq 10`

Since x ∈ Z,

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

(b)

We need to find the value of | x ∈ Z | or |A|. It means have to find the number of elements in set A.

`|A|=10`

| x ∈ Z | = 10

(c)

B = 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 > 10, x² ≤ 100| or |B|. It means have to find the number of elements in set B.

`|B|=0`

|x ∈ Z | = 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.