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### Interval as Intersection of Sets

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Date: 09/04/97 at 19:42:00
From: Jose M Clase
Subject: Algebra

- 4 is less than or equal to  y  is greater than or equal to 6

Describe the interval as the intersection of two other sets.
```

```
Date: 10/10/97 at 15:28:48
From: Doctor Chita
Subject: Re: Algebra

Hi Jose,

Inequalities are always a little tricky. I read your question to mean
that -4 <= y or y >=6.

One way to think about this inequality is to draw a number line,
putting a filled circle at -4 and at 6, since in each case y is equal
to both numbers. In the graph below, I used vertical line segments to
represent the filled circles at the two numbers.

<===============|---------------|==================>
-4              6

On the graph you can see that the numbers that satisfy your inequality
lie either to the left of -4 OR to the right of 6, and include -4
and 6. A number that is less than -4, such as -10, cannot also be
greater than 6. Similarly, a number that is greater than 6, like 26,
cannot also be less than -4. Therefore, your inequality represents
the union of two sets:

(1) the set of numbers such that -4 <= y or
(2) the set of numbers such that y >=6.

The way the problem is written -4 <= y or y >= 6 reflects the fact
that the two sets of numbers do not overlap.

On the other hand, the interval between -4 and 6 would represent
numbers that satisfy the inequality -4 < y < 6. The way this
inequality is written indicates that the values of y that satisfy
the inequality lie between the two endpoints, -4 and 6, but do not
include -4 and 6.

This set of numbers is the intersection of two sets: the set of
numbers that are greater than -4, like 0, and the set of numbers that
are less than 6, such as 0. Since there is a set of numbers that
satisfy both parts of the inequality  -4 < y < 6, the overlapping
region is called an intersection.

The intersection -4 < y < 6 is the "complement" of -4 <= y or y >= 6.
Together, the numbers that satisfy the union and the numbers that
satisfy the intersection make up the whole line.

If this explanation doesn't answer your question, write back to us and
we'll try again.

-Doctor Chita,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Sets

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