Let U = {1,2,3,4,5}, A = {1,2,3} and B = {2,3,4}. 1. Describe the relations between each of these sets. Use your own words. 2. Describe the relations between these sets using mathematical notation (statements!) 3. We wish to construct a new set S = An B. What is S? (Hint: Aºis the complement of A). 4. Draw a Venn Diagram highlighting S. 5. How many elements does P ({U}) have? Explain why.

The set A and B are subsets of set U. They're not complement of each other. The set S is {2,3}. The needed venn diagram is attached below. The size of power set of {U} is 2.

What is the number of elements in a power set?

Suppose the considered set is S

Then, its power set, denoted by P(S), is a set which contains all the possible subsets of S.

If S contains k elements, then symbolically we write [tex]n(S) = k[/tex]

Then the number of elements that the power set contains is:

[tex]n(P(S)) = 2^k[/tex]

What is a complement of a set?

Firstly there is a universal set, from which all the remaining sets of the context are constructed. We usually denote universal set by U.

Now let there is a set A.

Then, the complement set of A is such that it contains all the elements of U which are not in A (so that when A and its complement are combined, they can form A) .

Also, A and its complement have no intersection.

For this case, we're given that:

U = {1,2,3,4,5}, A = {1,2,3} and B = {2,3,4}, and S = A ∩ B

Evaluating each option one by one:

1: Describe the relations between each of these sets.

U is universal set, as it is a set which is super set (set containing all the elements of other set) of all the sets being discussed here.
A and B are subsets of U (subsets are sets which are formed by some or all or none elements of the set they're subset of.
A and B have two elements common which are 2 and 3, so they're not complement of each other.

2. Describe the relations between these sets using mathematical notation (statements!)

[tex]A \subset U\\B \subset U\\A \cap B \neq \phi[/tex]

([tex]\phi[/tex] is empty set, and the last statement says that intersection of A and B isn't empty set, so their intersection is some non empty set).

3. We wish to construct a new set S = A ∩ B. What is S?

Intersection of two sets is a set containing common elements of both the considered sets.

In A and B, two elements 2 and 3 are common.

Thus, S = A ∩ B = {2,3}

4. Draw a Venn Diagram highlighting S

The diagram is attached below.

5. How many elements does P ({U}) have?

Focusing on the set {U} will tell us that {U} is different than U

U is {1,2,3,4,5}

But {U} is { {1,2,3,4,5} }.

{U} is a set which contains a single element U. (An element of a set can be a set too).

Thus, P( {U} ) = [tex]2^1 = 2[/tex] (as size of {U} is 1, as it contains single element).

We have P( {U} ) = { { }, U} (the first element is empty set, the second element is U itself. Both are the subsets of {U}.

Thus, the set A and B are subsets of set U. They're not complement of each other. The set S is {2,3}. The needed venn diagram is attached below. The size of power set of {U} is 2.

Learn more about sets here:

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