Interval as Intersection of Sets

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/