Compound Inequalities

Union of Sets

The union of two sets, denoted A∪B ("A union B"), is the set of all members contained in either A or B or both. We can think of the union of two sets as the entire Venn diagram:

In our first example, A∪B = {0, 1, 2, 3, 4, 5, 6}:

In our second example, A∪B = { -10, -2, 0,, 4, 11, 15}:

Example 3: What is X∪Y if X = { -4, 3, 2, 11, -6} and Y = {3, 6, 11, -4, 5}?

The easiest way to write the union of two sets is to write all the members in the first set, and then write all the members in the second set that haven't been written yet:
X∪Y = { -4, 3, 2, 11, -6, 6, 5}

Example 4: What is A∪B if A = {2, 3, 4} and B = {5, 6, 7}?

A∪B = {2, 3, 4, 5, 6, 7}

The number of members in A∪B should be the total number of members in A plus the total number of members in B minus the number of members which are in both sets. In example 3, X contains 5 members, Y contains 5 members, and there are 3 members which are in both X and Y. Thus, X∪Y has 5 + 5 - 3 = 7 members.

Intersection of Sets

The intersection of two sets, denoted A∩B ("A intersect B") is the set of all members contained in bothA and B. We can think of the intersection of two sets as the overlap in the Venn diagram:

In our first example, A∩B = {2, 4, 6}:

In our second example, A∩B = :

Example 3. What is X∩Y if X = { -4, 3, 2, 11, -6} and Y = {3, 6, 11, -4, 5}?
X∩Y = { -4, 3, 11}
Example 4: What is A∩B if A = {2, 3, 4} and B = {5, 6, 7}?

Since A and B have no members in common, A∩B = ò (the null set).