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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:

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

  2. 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:

  1. A ∪ ∅ = A
  2. A ⋂ U = A
2.

Test the two equalities:

  1. A ∪ U = U
  2. A ⋂ ∅ = ∅

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