Basics Sets and set operations Definitions and Interval Notation Union and Intersection Cross product The Completeness Axiom Functions Definition and Notation How to find the Domain Evaluation Intercepts and Asymptotes Additional Resources 1 Sets and Set Operations A set is a collection of objects Sets must be defined in such a way that a reasonable observer can tell if any object is in the set or not this is a feature that mathematicians call well defined We will use symbols to talk about sets Sets use curly brackets to begin the description or list of elements and to end it A vertical bar in set brackets is read such that EXAMPLE 1 N is the set of all natural numbers x such that x is greater than 5 N x x is a natural number and x 5 This set is well defined You know that things like my car your shoe and the number 1 are not in the set while 100 7 and 58 are in the set Sets are unordered and never have duplicate elements EXAMPLE 2 a b c b c a a a c c c b The last set is really only 3 elements the repeated a s and b s are irrelevant There s only 3 elements in the last set Note that a b is not equal to b a Parenthesis imply order Nor does a a a mean the same thing as a a one is 3 dimensional and the other is 2 dimensional We will deal almost exclusively with sets of numbers in this class so a brief review of interval notation is probably in order See the handout on my website for a more complete review of interval notation Interval notation is a way to indicate a set of numbers that is a piece of the number line in an efficient in text manner Sets that are pieces of the number line have boundary numbers that may or may not be included in the set If a boundary is not included you indicate this by using a parenthesis symbol adjacent to the number if a boundary is included you use a square bracket adjacent to the number The symbol for infinity whether negative or positive always gets a parenthesis symbol 2 EXAMPLE 3 The set of all numbers greater than negative 5 can be written in the following four ways x x 5 and x is a real number set builder noation or graphically or x 5 or 5 5 EXAMPLE 4 x 2 x 10 says The set of all numbers between 2 and 10 including 2 but not 10 In other ways 2 10 2 x 10 2 10 Note that 2 10 means only the two points on the number line and none of the points in between while 2 10 is ambiguous without the first sentence is it a point pair or an interval Depends Sets can be combined in many ways We ll stick to unions intersections and cross products Given two sets A and B the union of them denoted A B is the set of elements that are in A in B or in both Given two sets A and B the intersection of them denoted A B is the set of elements that are in both A and in B simultaneously 3 EXAMPLE 5 A x x 5 B x 2 x 5 C x x 8 D x x 2 A B x x 2 Sketch the graphs of the sets and their union A C x 5 x 8 Sketch these graphs as well A B The empty set these sets are disjoint Show this A D x x 2 x x 5 Sketch the graph of this union The cross product of two sets A and B denoted A B results in a set of ordered pairs with the first coordinate from A and the second from B 4 EXAMPLE 6 A 1 2 3 and B 0 1 A B 1 0 2 0 3 0 1 1 2 1 3 1 The resulting set is an unordered set with elements that are ordered pairs It is VERY different from the original two sets They have whole numbers as set elements Sketch this resultant set note that if A and B are finite sets that is you can count the set elements and arrive at a number then the number of pairs in the cross product is the product of the numbers of set elements in each set 5 The Completeness Axiom If a non empty set A has an upper bound then it has a least upper bound Clearly some definitions and pictures are needed Upper bound an upper bound is a number that is greater than or equal to any element in the set bound is short for boundary Least upper bound The smallest of the upper bounds One logical extension of the axiom is that if there s a lower bound then there s a greatest lower bound for a non empty set of numbers What would be good definitions for lower bound and greatest lower bound EXAMPLE 7 a set with an upper and lower bound Suppose you have 3 7 8 is an upper bound and 7 is the least upper bound 1 3 7 8 For the example above 1 is a lower bound and 3 is the greatest lower bound Note that 7 is a set element and 3 is not The axiom and it s extension do not address this issue directly and by this omission allow either situation to happen 6 EXAMPLE 8 an unbounded set As a matter of best practices especially in a math class it is often necessary to give an example of what is NOT being defined to demonstrate that the new concept is necessary 3x 1 x is an integer 8 5 2 1 4 7 10 13 This set goes infinitely far in both the negative and the positive direction EXAMPLE 9 a set with two bounds a glb and a lub x x 1 1 where n is a natural number n Where are the bounds Let s begin by making a partial list of set elements n 1 n 1 1 0 2 1 2 1 2 3 1 3 2 3 4 1 4 3 4 5 1 5 4 5 1 1 n Now put the set elements on a number line so we can see where they re headed Let s talk about the bounds for this set EXAMPLE 10 another set with a glb and an lub 7 x x n 1 where n is a natural number What are the natural numbers What does an exponent of 1 mean Make a list of set elements graph them on a number line discuss lower bounds and upper bounds Make a number line sketch of this set EXAMPLE 11 A set that is bounded below but unbounded above x x p 1 where p is a real number and 0 p 1 What are the p s What are their reciprocals Do a 2 dimensional graph of …