ACCELERATED PRE-CALCULUS SUMMER STUDY GUIDE

CHAPTER 12: SETS, RELATIONS AND FUNCTIONS

SECTION 12.1: SETS

A "set" is a collection of "objects." Each object in the set is called an "element of the set." The elements usually have something in common.

For example, the following, {monkey, bear, lion, giraffe, zebra}, is called a set of elements. What do these elements have in common? One answer might be that they are animals in a local zoo.

Notice the symbols "{" and "}". We will call theses symbols "set enclosures."   "{" is the "left set enclosure" and "}" is the "right set enclosure."

The following set {2, 4, 6, 8, 10, ...} is a collection of objects that are numbers. The elements are 2, 4, 6, 8 and 10, and what these elements have in common is that they are positive even numbers. Notice that the set {2, 4, 6, 8, 10, ...} has 3 periods after the number 10. These three periods, called "ellipsis," mean “and so forth” or suggest that the elements in a set continue on forever or as mathematicians say "continue on to infinity."

When we need to represent some "recognizable pattern" mathematicans often use ellipsis. Consider the following set: {3, 6, 9,..., 90, 93, 96}. I have left out many numbers from this set but the ellipses suggest I am incrementing each subsequent number by 3 and the set contains all the numbers incremented by 3, starting at 3, going to the number 96.

Can you see the pattern in the following? {...1, 2, 4, 8, 16, 32, 64, 128...} If you take any number in this set and multiply it by 2 you will get the next number in the set. Pattern recognition skills are very important in the world of mathematics!

SECTION 12.2: DOMAIN, RANGE, RELATION

Let's look at this set: { (1,3), (-2,4), (5,-2), (-1,-3) } This is a set of ordered pairs which can be used to represent points in the Cartesian Coordinate System. The points below can be represented by our set of ordered pairs { (1,3), (-2,4), (5,-2), (-1,-3) }.

The set of x-values of a set of ordered pairs { (1,3), (-2,4), (5,-2), (-1,-3) } is named the "domain." The domain in this instance is {-2, 1, -1, 5}.

The set of y-values of a set of ordered pairs is named the "range." The range in this instance is {4, 3, -2, -3}.

The set of ordered pairs is also named a "relation." So { (1,3), (-2,4), (5,-2), (-1,-3) } is a relation.

EXAMPLES

ex. 1) State the domain and range of the relation {(5, –3), (4, 6), (3, –1), (6, 6), (5, 3)}. 

ans: The domain is {5, 4, 3, 6} and the range is {–3, 6, –1, 3}
Comment: Notice that the x coordinate, 5, occurs twice in {(5, –3), (4, 6), (3, –1), (6, 6), (5, 3)}. Mathematicians have agreed to write an x value in the domain once even though it may occur more than once in the relation. The same goes for the range: notice the y coordinate 6 occurs twice in the relation but is written only once in the range.

ex. 2) State the domain and range of the relation {(–3, 4), (–1, 4), (0, 4), (1, 4), (2, 4)}.

ans: The domain is {-3, -1, 0, 1, 2} and the range is {4}

SECTION 12.3: FUNCTIONS

There is a very special "relation" called a "function." Functions are very powerful in the world of applications as we will see in Chapter 16 of this manual.

12.4 GRAPHS OF FUNCTIONS VERSES NON-FUNCTIONS

A set of ordered pairs that is a function will have more than one point falling on a vertical line. Let's compare the graph of a function to a graph of a non function.

FUNCTION		NOT A FUNCTION
{ (1,3), (-2,4), (5,-2), (-1,-3) }		{ (1,3), (1, 5), (-2,4), (5,-2), (-1,-3) }

FUNCTION		NOT A FUNCTION
{ (7,-2), (-2,-2), (4,-2), (-3,-2) }		{ (7,5), (-7, 5), (-7,4), (-7,-2), (6,-3) }

We will be looking at functions in greater depth in chapter 16. This was a brief introduction.