Algebra Know-It-ALL

U as the set of all the points on or inside the oval. When you have two overlapping sets, one

of them can be a subset of the other, but this does not have to be the case. It is clearly not true

of the two sets T and U in Fig. 2-3. Neither of these sets is a subset of the other, because both

have some elements all their own.

Set Intersection

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.

Set Intersection 27