Lecture 1: Sets and Real Numbers : Lecture 1: Sets and Real Numbers Josh C. Shott
Objectives : Objectives Define sets and their basic properties
Define the subsets that comprise the set of real numbers Algebra Lecture 1 by Josh C. Shott
Sets : Sets set member/element The objects in a set are referred to as members.
(usually denoted by lowercase letters) Examples A = { red, yellow, blue, green } B = { ferrari, corvette, porsche, masarati } C = { 1, 2, 3, 4 } D = { 2, 4, 6, 8, … } ellipsis A group of objects
(usually denoted by capital letters (S)) Algebra Lecture 1 by Josh C. Shott
Sets : Sets equality every object in set X is also found in set Y. membership x is a member of the set X Examples Is X = Y a true statement? Yes Algebra Lecture 1 by Josh C. Shott
Sets : What is ? Sets subsets every object in set X is also found in set Y, however there can be more objects in Y. union The union of sets X and Y is the set of all elements that are in X and all the elements in Y Examples Is a true statement? Yes Algebra Lecture 1 by Josh C. Shott
Sets : What is ? Sets intersection the set of elements that are common to both the set X and Y. set-builder notation set builder notation is used to describe a set in conditions where listing all the elements of the set is not practical or possible Example dummy variable such that condition that describes X Algebra Lecture 1 by Josh C. Shott
Practice Examples : Practice Examples Denote the set that contains all the different values of common U. S. coins.
Use set-builder notation to describe a set that contains all the numbers that are less than 6.
Given: A = {1} and B = {o, n, e}. Is A = B a true statement?
If X = {t,o}, Y = {t,w,o}, Z = {t,o,o,t}, W = {t,o,w}, and V = {t,o,o}, which of the following sets are equal?
Given the sets in Problem 4, what is and ?
If X = {t,o} and Y = {t,w,o}, is ? Algebra Lecture 1 by Josh C. Shott
Practice Examples : Practice Examples Denote the set that contains all the different values of common U. S. coins. Use set-builder notation to describe a set that contains all the numbers that are less than 6. Given: A = {1} and B = {o, n, e}. Is A = B a true statement? If X = {t,o}, Y = {t,w,o}, Z = {t,o,o,t}, W = {t,o,w}, and V = {t,o,o}, which of the following sets are equal? X = {t,o} V = {t,o,o} Z = {t,o,o,t} Y = {t,w,o} W = {t,o,w} Algebra Lecture 1 by Josh C. Shott
Practice Examples : Practice Examples V = {t,o,o} W = {t,o,w} Given the sets in Problem 4, what is and ? If X = {t,o} and Y = {t,w,o}, is ? Every object in Y is not found in X, therefore However, every object in X is found in Y, therefore Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System closure property Natural numbers (N) The natural numbers are the numbers that come natural to everyone. The most natural thing to do with numbers is count objects. Thus the natural numbers are also referred to as the counting numbers. A set has the closure property when an arithmetic operation (addition, subtraction, mutliplication, division) performed on two elements in a set results in the same set.
(Examples will follow) The natural numbers are closed under addition and multiplication.
Explanation: If you add any two natural numbers, the result is a natural number. If you multiply any two natural numbers, the result is a natural number. This is the closure property. Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System Integers (Z) The closure property separates the natural numbers from the integers, in that the integers are also closed by subtraction. The set of integers are the natural numbers, their opposites, and 0. All of the natural numbers are integers, thus The integers are closed under addition, subtraction and multiplication. Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System Rational Numbers (Q) The closure property separates the rational numbers from the integers, in that the rational numbers are also closed by division The rational numbers is the set of numbers formed from the quotient of two integers. All the integers are also rational numbers, thus The rational numbers are closed under addition, subtraction, multiplication, and division (with the exception of 0) terminating digits repeating digits Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System Irrational Numbers (I) The closure property does not apply to the irrational numbers. The irrational numbers are not closed under any arithmetic operation. The irrational numbers is the set of numbers that cannot be expressed as the quotient of two integers. Irrational numbers are not rational numbers, thus The digits of irrational numbers do not terminate and do not infinitely repeat a pattern of the same digits. Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System Irrational Numbers (I) The following are some counter examples of the closure property involving irrational numbers: multiplication division addition subtraction Algebra Lecture 1 by Josh C. Shott
The Real Number System : The Real Number System Real Numbers (R) Question: Are the real numbers closed under any arithmetic operations? Answer: Yes! The real numbers are closed under all 4 operations, with the exception of 0. Algebra Lecture 1 by Josh C. Shott
Practice Examples : Practice Examples Given the set
Find the following subsets
natural numbers
integers
rational numbers
irrational numbers Algebra Lecture 1 by Josh C. Shott
Slide 17 : ? Thank you for your time Algebra Lecture 1 by Josh C. Shott