Associated Topics || Dr. Math Home || Search Dr. Math

Unions and Intersections: Proving Sets

```
Date: 10/17/1999 at 18:33:04
From: Don Edgar
Subject: Proving sets with unions and intersections

The question given to me is: A student is asked to prove that for any
sets A, B and C; A - (B union C) = (A - B) intersect (A - C). The
student writes 'Let x be an element of A - (B union C). Then x is an
element of A and x is not an element of B, or x is not an element of
C. Therefore, x is an element of A - B and x is an element of A - C.
Thus, A - (B union C) = (A - B) intersect (A - C).' What, if anything,
is wrong with this proof?

I have no idea what to do or how to go about this. I'm not sure if
he's right or not or how he even came up with that thinking. If you
could help me I would be most appreciative.

Thank you very much.
DE
```

```
Date: 10/17/1999 at 22:06:06
From: Doctor Ian
Subject: Re: Proving sets with unions and intersections

Hi Don,

I'm going to reformat the proof so I can refer to specific parts of
it:

[1] Let x be an element of A-(B union C).

[2] Then x is an element of A
and x is not an element of B
or x is not an element of C.

[3] Therefore, x is an element of A - B
and x is an element of A - C.

[4] Thus, A-(B union C) = (A - B)intersect(A - C).

Technically, the third line of [2] should begin with 'and' instead of
'or'. Other than that, it's correct.

It's easier to see what's going on with pictures. And the pictures are
a lot better looking if you use overlapping circles instead of
rectangles, but I have to do this with a keyboard, so...

Imagine three sets - A, B and C - represented by three overlapping
rectangles.

+---------------+
|            B  |
+--------|----+          |      (Figure 1)
| A      |    |          |
|        |    |          |
|   +--------------+     |
|   |    |    |    |     |
|   |    +----|----------+
|   |         |    |
+---|---------+    |
|              |
|      C       |
+--------------+

The union of B and C is everything that is in B, or in C, or both:

+---------------+
|XXXXXXXXXXXXXXX|
+--------|----+XXXXXXXXXX|      (Figure 2)
| A      |XXXX|XXXXXXXXXX|
|        |XXXX|XXXXXXXXXX|
|   +--------------+XXXXX|
|   |XXXX|XXXX|XXXX|XXXXX|
|   |XXXX+----|----------+
|   |XXXXXXXXX|XXXX|
+---|---------+XXXX|
|XXXXXXXXXXXXXX|
|XXXXXXXXXXXXXX|
+--------------+

The part left unfilled in the diagram above is everything that is in A
that isn't in the union of B and C -- in other words, A - (B union C).

A - B is everything that is in A that isn't also in B:

+---------------+
|            B  |
+--------|----+          |      (Figure 3)
|XXXXXXXX|    |          |
|XXXXXXXX|    |          |
|XXX+--------------+     |
|XXX|XXXX|    |    |     |
|XXX|XXXX+----|----------+
|XXX|XXXXXXXXX|    |
+---|---------+    |
|              |
|      C       |
+--------------+

A - C is everything that is in A that isn't also in C:

+---------------+
|            B  |
+--------|----+          |      (Figure 4)
|XXXXXXXX|XXXX|          |
|XXXXXXXX|XXXX|          |
|XXX+--------------+     |
|XXX|    |    |    |     |
|XXX|    +----|----------+
|XXX|         |    |
+---|---------+    |
|              |
|      C       |
+--------------+

The intersection of these two sets is the part they have in common,

+---------------+
|            B  |
+--------|----+          |      (Figure 5)
|XXXXXXXX|    |          |
|XXXXXXXX|    |          |
|XXX+--------------+     |
|XXX|    |    |    |     |
|XXX|    +----|----------+
|XXX|         |    |
+---|---------+    |
|              |
|      C       |
+--------------+

The part that is left unfilled in Figure 2 is A - (B union C).

The part that is filled in Figure 5 is (A - B) intersect (A - C).

They're the same. So A - (B union C) = (A - B) intersect (A - C).

Now let's look at the (corrected) proof again:

[1] LET x BE AN ELEMENT OF A - (B UNION C).

This is given.

[2] THEN x IS AN ELEMENT OF A

Because it's in the part of A that isn't in some other set,
so it must at least be in A.

AND X IS NOT AN ELEMENT OF B

Because if it were an element of B, it would be in
B union C.

AND X IS NOT AN ELEMENT OF C.

Because if it were an element of C, it would be in
B union C.

[3] THEREFORE X IS AN ELEMENT OF A - B

Because it's in A, but not in B.

AND X IS AN ELEMENT OF A - C.

Because it's in A, but not in C.

[4] THUS, A - (B UNION C) = (A - B) INTERSECT (A - C).

Because it's in both (A - B) and in (A - C).

I hope this helps. Be sure to write back if you're still confused, or
if you have any other questions.

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search