Theintersection of two sets is made up of all elements that belong to both of the sets. When you have two sets, say V and W, their intersection is also a set, and it is written V∩W. The upside-down U-like symbol is read “intersect,” so you would say “V intersect W.”
Intersection of two congruent sets
When two nonempty sets are congruent, their intersection is the set of all elements in either set. You can write it like this, for any nonempty sets X and Y,
If X=Y then X∩Y=X
and X∩Y=Y
But really, you’re dealing with only one set here, not two! So you could just as well write X∩X=X
This also holds true for the null set:
∅ ∩ ∅ = ∅
Intersection with the null set
The intersection of the null set with any nonempty set gives you the null set. This fact is not so trivial. You might have to think awhile to fully understand it. For any nonempty set V, you can write
V∩ ∅ = ∅
Remember, any element in the intersection of two sets has to belong to both of those sets. But nothing can belong to a set that contains no elements! Therefore, nothing can belong to the intersection of the null set with any other set.
Intersection of two nonempty disjoint sets
When two nonempty sets are disjoint, they have no elements in common, so it’s impossible for anything to belong to them both. The intersection of two disjoint sets is always the null set. It doesn’t matter how big or small the sets are. Remember the sets of even and odd whole numbers,Weven and Wodd? They’re both infinite, but
Weven∩Wodd= ∅
Intersection of two overlapping sets
When two sets overlap, their intersection contains at least one element. There is no limit to how many elements the intersection of two sets can have. The only requirement is that every element in the intersection set must belong to both of the original sets.
Let’s look at the examples of overlapping sets you saw a little while ago, and figure out the intersection sets. First, examine these
L= {2, 3, 4, 5, 6} M= {6, 7, 8, 9, 10} Here, the intersection set contains one element:
L∩M= {6}
That means the set containing the number 6, not just the number 6 itself. Now look at these: P= {21, 23, 25, 27, 29, 31, 33}
Q= {25, 27, 29, 31, 33, 35, 37} The intersection set in this case contains five elements:
P∩Q= {25, 27, 29, 31, 33} Now check these sets out:
R= {11, 12, 13, 14, 15, 16, 17, 18, 19} S= {12, 13, 14}
In this situation, S⊂R, so the intersection set is the same as S. We can write that down as follows: R∩S=S
= {12, 13, 14}
How about the set W3− of all positive, negative, or zero whole numbers less than or equal to 3,
and the set W0+ of all the nonnegative whole numbers?
W3−= {..., −5,−4,−3,−2,−1, 0, 1, 2, 3}
W0+= {0, 1, 2, 3, 4, 5, ...}
Here, the intersection set has four elements:
W3−∩W0+= {0, 1, 2, 3}
Figure 2-4 is a Venn diagram that shows two overlapping sets. Think of V as the rectangle and everything inside it. Imagine W as the oval and everything inside it. The two regions are hatched diagonally, but in different directions. The intersection V∩ W shows up as a double-hatched region.
Are you confused?
Go back and look again at Fig. 2-1. You can see that the set of all women in Chicago (call it Cw) is a proper subset of the set of all people in the state of Illinois (call it Ip). You would write down this fact as follows:
Cw⊂Ip
The diagram also makes it clear that the intersection of set Cw with set Ip is just the set Cw. In order to be in both sets, a person must be a woman in Chicago, that is, an element of Cw. Here’s how you would write that
Cw∩Ip=Cw
You can always draw a Venn diagram if it will help you understand how sets are related.
Here’s a challenge!
Find two sets of whole numbers that overlap, with neither set being a subset of the other, and whose inter- section set contains infinitely many elements.
Set Intersection 29 Universe
V
W VUW
Figure 2-4 Two overlapping sets, V and W. Their intersection is shown by the double-hatched region.
Solution
There are countless examples of set pairs like this. Let’s look at the set of all positive whole numbers divis- ible by 4 without a remainder. (When there is no remainder, a quotient comes out as a whole number.) Name this set W4d. Similarly, name the set of all positive whole numbers divisible by 6 without a remainder
W6d. Then
W4d= {0, 4, 8, 12, 16, 20, 24, 28, 32, 36, ...}
W6d= {0, 6, 12, 18, 24, 30, 36, 42, 48, ...}
Both of these sets have infinitely many elements. They overlap, because they share certain elements. But neither is a subset of the other, because they both have some elements all their own. Their intersection is the set of elements divisible by both 4 and 6. Let’s call it W4d6d. If you’re willing to write out both of the above lists up to all values less than or equal to 100, you will see that
W4d∩W6d=W4d6d
= {0, 12, 24, 36, 48, 60, 72, 84, 96, ...}
This is an infinite set, and it happens to be the set of all positive whole numbers divisible by 12 without a remainder (call it W12d). We can write
W4d∩W6d=W12d
Set Union
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
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}
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.