Question

N = {natural numbers}. Check all correct statements about the power set 2ᴺ.

a. {0} is an element of 2ᴺ
b. Z = {integers) is a subset of 2ᴺ
c. 2ᴺ contains the set of natural numbers N as an element
d. 2ᴺ is a subset of 2ᴺ
e. {1},{1,2}, {1,2,3}, {1,2,3,4} are some of the elements of 2ᴺ
f. Cardinality of 2N is finite
g. 2ᴺ is infinite
h. 2ᴺ contains empty set as an element
i. {even natural numbers) are an element of 2ᴺ
j. 2ᴺ is a subset of R= {real numbers)
k. 2ᴺ is not well-defined

Answer 1

The power set 2ᴺ includes all possible subsets of a given set, and the statements a, d, f, h are correct about the power set.

Step-by-step explanation:

To check which statements about the power set 2ᴺ are correct:

a. {0} is an element of 2ᴺ - This statement is correct because the power set includes all possible subsets, and one of the subsets can be the set containing only the element 0.

b. Z = {integers} is a subset of 2ᴺ - This statement is incorrect because the power set 2ᴺ only includes subsets of the natural numbers, not the integers.

c. 2ᴺ contains the set of natural numbers N as an element - This statement is incorrect because the power set includes subsets made up of the natural numbers, but it does not include the entire set of natural numbers as a single element.

d. 2ᴺ is a subset of 2ᴺ - This statement is correct since the power set is a set of all subsets of a given set, including itself.

e. {1}, {1,2}, {1,2,3}, {1,2,3,4} are some of the elements of 2ᴺ - This statement is incorrect because these sets are subsets of the natural numbers, but they are not elements of the power set 2ᴺ.

f. Cardinality of 2N is finite - This statement is correct because the power set 2ᴺ, being the set of all subsets of N, has a finite cardinality.

g. 2ᴺ is infinite - This statement is incorrect because while the power set 2ᴺ is large, it is not infinite.

h. 2ᴺ contains empty set as an element - This statement is correct because the empty set is a subset of every set, including the power set 2ᴺ.

i. {even natural numbers} are an element of 2ᴺ - This statement is incorrect because the power set 2ᴺ includes subsets made up of natural numbers, not individual elements like {even natural numbers}.

j. 2ᴺ is a subset of R = {real numbers} - This statement is incorrect because the power set 2ᴺ is made up of subsets of the natural numbers, not real numbers.

k. 2ᴺ is not well-defined - This statement is incorrect because the power set 2ᴺ is a well-defined concept in set theory.

Answer 2

The correct statements are:

a. {0} is an element of 2ᴺ.

d. 2ᴺ is a subset of 2ᴺ.

e. {1},{1,2}, {1,2,3}, {1,2,3,4} are some of the elements of 2ᴺ.

g. 2ᴺ is infinite.

h. 2ᴺ contains the empty set as an element.

i. {even natural numbers) are an element of 2ᴺ.

Step-by-step explanation:

The power set of a set (N), denoted as
`\(2^N\)`, is the set of all possible subsets of (N). Let's evaluate each statement:

a. True - {0} is a subset of (N), so it belongs to
`\(2^N\)`.

b. False - The set of integers (Z) is not a subset of (N), so it's not in
`\(2^N\)`.

c. False -
`\(2^N\)` consists of subsets of (N), not the set (N) itself.

d. True - Every set is a subset of itself, so
`\(2^N\)` is a subset of
`\(2^N\)`.

e. True - These are indeed subsets of (N) and therefore part of
`\(2^N\)`.

f. False - The cardinality of
`\(2^N\)` is
`\(2^(|N|)\)`, which is infinite for
`\(N = \{1, 2, 3, ...\}\)`.

g. True - As stated in (f), the power set is infinite.

h. True - The empty set is always an element of any power set.

i. True - The set of even natural numbers is a subset of (N) and thus in
`\(2^N\)`.

j. False -
`\(2^N\)` is not a subset of the real numbers (R); it contains subsets of (N).

k. False - The power set is well-defined and consists of all subsets of a given set.

In summary, statements a, d, e, g, h, and i are true, while the rest are false based on the properties of power sets and subsets.