One operation that is frequently used to form new sets from old ones is called the union. In common usage the word union signifies a bringing together, such as unions in organized labor or the State of the Union address that the U.S. President makes before a joint session of Congress. In the mathematical sense, the union of two sets retains this idea of bringing together. More precisely, the union of two sets A and B is the set of all elements x such that x is an elements of the set A or x is an element of the set B.
The word that signifies that we are using a union is the word "or."
The Word Or
When we use the word "or" in day to day conversations, we may not realize that this word is being used in two different ways. The way is usually inferred from the context of the conversation. If you were asked “Would you like the chicken or the steak?” the usual implication is that you may have one or the other, but not both. Contrast this with the question, “Would you like butter or sour cream on your baked potato?” Here "or" is used in the inclusive sense in that you could choose only butter, only sour cream, or both butter and sour cream.
In mathematics the word "or" is used in the inclusive sense. So the statement, "x is an element of A or an element of B" means that one of the three are possible:
- x is an element of just A and not an element of B
- x is an element of just B and not an element of A.
- x is an element of both A and B. (We could also say that x is an element of the intersection of A and B
An Example
For an example of how the union of two sets forms a new set, let’s consider the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. To find the union of these two sets, we simply list every element that we see, being careful not to duplicate any elements. The numbers 1, 2, 3, 4, 5, 6, 7, 8 are in either one set or the other, therefore the union of A and B is {1, 2, 3, 4, 5, 6, 7, 8 }.
Notation for Union
In addition to understanding the concepts concerning set theory operations, it is important to be able to read symbols used to denote these operations. The symbol used for the union of the two sets A and B is given by A ∪ B. One way to remember the symbol ∪ refers to union is to notice its resemblance to a capital U, which is short for the word “union.” Be careful, because the symbol for union is very similar to the symbol for intersection. One is obtained from the other by a vertical flip.
To see this notation in action, refer back the above example. Here we had the sets A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7, 8}. So we would write the set equation A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8 }.
Union With the Empty Set
One basic identity that involves the union shows us what happens when we take the union of any set with the empty set, denoted by #8709. The empty set is the set with no elements. So joining this to any other set will have no effect. In other words, the union of any set with the empty set will give us the original set back
This identity becomes even more compact with the use of our notation. We have the identity: A ∪ ∅ = A.
Union With the Universal Set
For the other extreme, what happens when we examine the union of a set with the universal set?
Since the universal set contains every element, we cannot add anything else to this. So the union or any set with the universal set is the universal set.
Again our notation help us to express this identity in a more compact format. For any set A and the universal set U, A ∪ U = U.
Other Identities Involving the Union
There are many more set identities that involve the use of the union operation. Of course it's always good to practice using the language of set theory. A few of the more important are stated below. For all sets A, and B and D we have:
- Reflexive Property: A ∪ A =A
- Commutative Property: A ∪ B = B ∪ A
- Associative Property: (A ∪ B) ∪ D =A ∪ (B ∪ D)
- DeMorgan’s Law I: (A ∩ B)^{C} = A^{C} ∪ B^{C}
- DeMorgan’s Law II: (A ∪ B)^{C} = A^{C} ∩ B^{C}