Algebra Know-It-ALL

32 The Language of Sets

Universe

XUY

X

Y

Figure 2-5 Two overlapping sets, X and Y. Their union is

shown by the entire shaded region.

That’s all the odd whole numbers between, and including, 21 and 37. We count the duplicate

elements 25 through 33 only once. Now look at these:

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

S= {12, 13, 14}

In this situation, S⊂R, so the union set is the same as R. We can write that down this way:

R∪S=R

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

We count the elements 12, 13, and 14 only once. Now these:

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

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

Here, the union set consists of all the positive and negative whole numbers, along with zero.

Let’s write that set as W 0 ± (read “W sub zero plus-or-minus”). Then

W 3 −∪W 0 +=W 0 ±

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

The elements 0, 1, 2, and 3 are counted only once. This set W 0 ± is usually called the set of

integers. We’ll work more with integers in the chapters to come.

Figure 2-5 is a Venn diagram that shows two overlapping sets. Think of X as the rectangle

and everything inside it. Imagine Y as the oval and everything inside it. The union of the sets,

X∪Y, is shown by the entire shaded region inside the outer solid line. Part of that line is the