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.

the answer is:

the answer is:

to solve the problem, we need to remember the quotient of power property, it's defined by the following relation:

if we have a quotienf of powers that have the same base, we need to keep the same base and subtract the exponent of the denominator power to the exponent of the numerator power.

so, we are given the expression:

then, calculating we have:

hence, the answer is:

have a nice day!

explanation:

step-by-step explanation: