Theunion of two sets contains all of the elements that belong to one set or the other, or both.
When you have two sets, say X and Y, their union is also a set, written X∪ Y. The U-like symbol is read “union,” so you would say “X union Y.”
Union of two congruent sets
When two nonempty sets are congruent, their union is the set of all elements in either set. For any nonempty sets X and Y,
If X=Y then X∪Y=X
and X∪Y=Y
But you’re really dealing with only one set here, so you could just as well write X∪X=X
30 The Language of Sets
And for the null set
∅ ∪ ∅ = ∅
When two sets are congruent, their union is the same as their intersection. This might seem trivial right now, but there are situations where it’s not clear that two sets are congruent. In cases like that, you can compare the union with the intersection as a sort of congruence test.
If the union and intersection turn out identical, then you know the two sets in question are congruent.
Union with the null set
The union of the null set with any nonempty set gives you that nonempty set. For any nonempty set X, you can write
X∪ ∅ =X
Remember, any element in the union of two sets only has to belong to one of them.
Union of two disjoint sets
When two nonempty sets are disjoint, they have no elements in common, but their union always contains some elements. Consider again the sets of even and odd whole numbers, Weven
andWodd. Their union is the set of all the whole numbers. So Weven∪Wodd= {0, 1, 2, 3, 4, 5, ...}
Union of two overlapping sets
Again, let’s look at the same examples of overlapping sets we checked out when we worked with intersection. First
L= {2, 3, 4, 5, 6}
M= {6, 7, 8, 9, 10}
The union set here contains nine elements:
L∪M= {2, 3, 4, 5, 6, 7, 8, 9, 10}
The number 6 appears in both sets, but we count it only once in the union. (An element can only “belong to a set once.”) Now look at these:
P= {21, 23, 25, 27, 29, 31, 33}
Q= {25, 27, 29, 31, 33, 35, 37}
The union set in this case is
P∪Q= {21, 23, 25, ..., 33, 35, 37}
Set Union 31
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:
W3−= {..., −5,−4,−3,−2,−1, 0, 1, 2, 3}
W0+= {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 W0± (read “W sub zero plus-or-minus”). Then W3−∪W0+=W0±
= {..., −5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5, ...}
The elements 0, 1, 2, and 3 are counted only once. This set W0± 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
outside of the rectangle and part of it is the outside of the oval. Any element inside the region bounded by the dashed line is counted only once.
Are you confused?
Once more, go back and look at Fig. 2-1, again noting that the set of all the women in Chicago is a proper subset of the set of all the people in Illinois, that is, Cw⊂Ip. The diagram also makes it plain that the union ofCw with Ip is just Ip. To be in one set or the other (or both), a person only has to be a resident of Illinois, that is, an element of Ip. It’s not necessary to be a woman, and it’s not necessary to be in Chicago. Here’s how you would write that:
Cw∪Ip=Ip
Here’s a challenge!
Can you find two sets of whole numbers, with one of them infinite, but such that their union contains only a finite number of elements?
Solution
Don’t think about this for too long. You’ll never find two such sets! An element in the union of two sets only has to belong to one of the sets. If a set has infinitely many elements, then the union of that set with any other set—even the null set—must have infinitely many elements as well.
Practice Exercises
This is an open-book quiz. You may (and should) refer to the text as you solve these problems.
Don’t hurry! You’ll find worked-out answers in App. A. The solutions in the appendix may not represent the only way a problem can be figured out. If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it!
1. Is there any set that is a subset of every other set? If so, what is it? If such a set can’t exist, why not?
2. Continuing with the theme of Problem 1, is there a way to take nothing and build up an unlimited number of different sets from it? If so, show an example. If not, explain why not.
3. What set does the small, dark-shaded triangle marked P represent in Fig. 2-6? What set does the dark-shaded, irregular, four-sided figure marked Q represent?
4. If you consider all the possible intersections of two sets in Fig. 2-6, which of those intersection sets are empty?
5. Is the universal set a subset of itself? Is it a proper subset of itself?
6. Give an example of two sets, both with infinitely many elements, but such that one is a proper subset of the other.
Practice Exercises 33
7. What is the intersection of these two sets? What is their union?
A= {1, 1/2, 1/3, 1/4, 1/5, 1/6, ...}
G= {1, 1/2, 1/4, 1/8, 1/16, 1/32, ...}
In set A, the denominator of the fraction increases by 1 as you go down the list. In set G, the denominator doubles as you go down the list. All the numerators in both sets are equal to 1.
8. List all the subsets of {1, 2, 3}. Here’s a hint: Whenever you want to find all the subsets of a small set like this, first list its individual elements. Then make up every possible set that contains at least one of those elements. Finally, be sure to include the empty set, which is a subset of any other set.
9. List all the subsets of {1, {2, 3}}. Be careful. The hint given with Problem 8 is important here.
10. List all the subsets of {1, {2, {3}}}. Be extra careful! The hint given with Problem 8 is even more important here.
34 The Language of Sets Universe
A
B
C
D E
P
Q
Figure 2-6 Illustration for Probs. 3 and 4.
The whole numbers starting with 0 and counting upward are usually called the natural num-bers. Some mathematicians don’t call 0 a natural number. It’s a little like the dispute among astronomers over whether Pluto should be called a planet, or whether empty space should be called a part of nature. In this book, we’ll call 0 a natural number.