Algebra Know-It-ALL

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

30 The Language of Sets