Algebra Know-It-ALL

Overlapping sets

When two sets have at least one element in common, they are called overlapping sets. In formal

texts you might see them called nondisjoint sets. Congruent sets overlap in the strongest pos-

sible sense, because they have all their elements in common. More often, two overlapping sets

share some, but not all, of their elements. Here are two sets of numbers that overlap with one

element in common:

L= {2, 3, 4, 5, 6}

M= {6, 7, 8, 9, 10}

Here is a pair of sets that overlap more:

P= {21, 23, 25, 27, 29, 31, 33}

Q= {25, 27, 29, 31, 33, 35, 37}

Technically, these sets overlap too:

R= {11, 12, 13, 14, 15, 16, 17, 18, 19}

S= {12, 13, 14}

Here, you can see that S is a subset of R. In fact, S is a proper subset of R. Now, let’s look at a

pair of infinite sets that overlap with four elements in common:

W 3 −= {..., −5,−4,−3,−2,−1, 0, 1, 2, 3}

W 0 += {0, 1, 2, 3, 4, 5, ...}

The notation W 3 − (read “W sub three-minus”) means the set of all positive or negative whole

numbers starting at 3 and decreasing, one by one, without end. The notation W0+ (read

“W sub zero-plus”) means the set of whole numbers starting with 0 and increasing, one by

one, without end. That’s the set of whole numbers as it is usually defined.

Figure 2-3 is a Venn diagram that shows two sets, T and U, with some elements in com-

mon, so they overlap. You can imagine T as the set of all the points on or inside the circle, and

26 The Language of Sets

Universe

T

U

Figure 2-3 Two overlapping sets, T and U. They

have some elements in common.