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/
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