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.

(a)

(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

               ... (1)

Taking square root on both sides.

            .... (2)

Using (1) and (2) we get

Since x ∈ Z,

(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.

| {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

               ... (3)

It means

            .... (4)

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

(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.

|{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

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

Therefore |P(A)|=1024.

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