The operations on sets satisfy many identities. We will look at some important ones. Note the close similarity between these properties and their corresponding properties for addition and multiplication.

#### Identity Property for Union:

The Identity Property for Union says that the union of a set and the empty set is the set itself. For a set A,

A ∪ ∅ = A.

Let A = {3, 5, 7, 11} and B = { }. Then A ∪ B = {3, 5, 7, 11} ∪ { } = {3, 5, 7, 11}.

The empty set is the identity element for the union of sets. What would be the identity element for the addition of whole numbers? It is the number 0.

#### Identity Property for Intersection:

The intersection of a set with the Universal set is the set itself. For a set A, A ⋂ U = A.

Let A = {3, 5, 7, 11} and N, the set of all natural numbers. Then A ⋂ N = {3, 5, 7, 11}.

The Universal set is the identity for the intersection of sets. What would be the identity for the multiplication of whole numbers? It is the number 1.

#### Domination Laws:

Any set intersected with an empty set gives the empty set. A ⋂ ∅ = ∅.

The union of any set with the Universal set is the Universal set. A ∪ U = U.

Use the laws of set operations to prove that (M ⋂ ∅) ∪ N = N

Step 1: M ⋂ ∅ = ∅ (by Domination Law for intersection).

Step 2: (M ⋂ ∅) ∪ N = ∅ ∪ N (replacing the value of (M ⋂ ∅) from Step 1).

Step 3: ∅ ∪ N = N (by Identity property of union)

Therefore, (M ⋂ ∅) ∪ N = N

### Instructions

**1.**

You have been provided with Python code that creates a set A and a universal set U. Use these sets to test the two equalities:

- A ∪ ∅ = A
- A ⋂ U = A

**2.**

Test the two equalities:

- A ∪ U = U
- A ⋂ ∅ = ∅