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 xvalues 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 yvalues 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.
WHAT IS A FUNCTION? 
If each xvalue of a relation has one unique yvalue assigned to it, then the relation is named a "function."
ex. 1) { (1,3), (1, 5), (2,4), (5,2), (1,3) } is not a function! Look at the set of ordered pairs to your left. Note the ordered pairs in blue. The xvalue 1 has two different yvalues, 3 and 5.
Hence, we say { (1,3), (1, 5), (2,4), (5,2), (1,3) } is not a function.
ex. 2) { (1,3), (2,4), (5,2), (1,3) } is called a function.
Why? Because each xvalue has only 1 y value.
ex. 3) { (7,5), (7, 5), (7,4), (7,2), (6,3) } is not a function. Look at the set of ordered pairs to your left. The xvalue 7 has three different yvalues 5,4,and 2.
Hence, we say { (7,5), (7, 5), (7,4), (7,2), (6,3) } is not a function.
ex. 4) { (7,2), (2,2), (4,2), (3,2) } is a function.
Why? Because each unique xvalue has only 1 yvalue.
COMMENT: All functions are relations but not all relations are functions. Remember, a relation is nothing more than a set of ordered pairs. 
12.4 GRAPHS OF FUNCTIONS VERSES NONFUNCTIONS
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.



THE SUMMER STUDY GUIDE
BY CHAPTERS

RETURN TO THE SUMMER STUDY GUIDE MAIN PAGE
 CHAPTER 1: THE NUMBER SYSTEM
 CHAPTER 2: ORDER OF OPERATIONS
 CHAPTER 3: VARIABLES, MONOMIALS,
BINOMIALS, TRINOMIALS, POLYNOMIALS,
COEFFICIENTS, TERMS AND LIKE TERMS
 CHAPTER 4: SIGNED NUMBERS,
ABSOLUTE VALUE, AND INEQUALITY SYMBOLS
 CHAPTER 5: FACTORS, COMMON
FACTORS, LEAST COMMON FACTORS AND GREATEST COMMON FACTORS
 CHAPTER 6: PROPERTIES OF NUMBERS
 CHAPTER 7: THE WORLD OF FRACTIONS
 CHAPTER 8: EXPONENTS
 CHAPTER 9: ROOTS
 CHAPTER 10: ALGEBRAIC EXPRESSIONS
 CHAPTER 11: CARTESIAN COORDINATE SYSTEM
 CHAPTER 12: SETS, RELATIONS AND FUNCTIONS
 CHAPTER 13: AVERAGE RATE OF CHANGE OF Y WITH RESPECT TO X, SLOPE, PYTHAGOREAN THEOREM, AND DISTANCE FORMULA BETWEEN TWO POINTS
 CHAPTER 14: XINTERCEPT(ZERO) AND Y INTERCEPT(B)
 CHAPTER 15: LINES
 CHAPTER 16: FUNCTIONS
 CHAPTER 17: MULTIPLYING POLYNOMIALS
 CHAPTER 18: FACTORING
 CHAPTER 19: RATIONAL EXPRESSIONS
 CHAPTER 20: SOLVING EQUATIONS
 CHAPTER 21:SOLVING INEQUALITIES
 CHAPTER 22: SOLVING A SYSTEM OF EQUATIONS
 CHAPTER 23: QUADRATICS
 CHAPTER 24: CIRCLES
 CHAPTER 25: AREAS AND PERIMETERS OF PLANE FIGURES
 CHAPTER 26: VOLUMES
