Lecture Discrete mathematics and its applications (7/e) – Chapter 2: Basic structures: sets, functions, sequences, sums, and matrices. In this chapter we will establish the surprising result that the set of rational numbers is countable, while the set of real numbers is not. This chapter will also show how the concepts we discuss can be used to show that there are functions that cannot be computed using a computer program in any programming language. | Basic Structures: Sets, Functions, Sequences, Sums, and Matrices Chapter 2 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences and Summations Types of Sequences Summation Formulae Set Cardinality Countable Sets Matrices Matrix Arithmetic Sets Section Section Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets Tuples Cartesian Product Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory is an important branch of mathematics. Many different systems of axioms have been used to develop set theory. Here we are not concerned with a formal set of axioms for set theory. Instead, we will use what is called naïve set theory. Sets A set is an unordered collection of objects. the students in this class the chairs in this room The objects in a set are called the elements, or members of the set. A set is said to contain its elements. The notation a ∈ A denotes that a is an element of the set A. If a is not a member of A, write a ∉ A Describing a Set: Roster Method S = {a,b,c,d} Order not important S = {a,b,c,d} = {b,c,a,d} Each distinct object is either a member or not; listing more than once does not change the set. S = {a,b,c,d} = {a,b,c,b,c,d} Elipses ( ) may be used to describe a set without listing all of the members when the pattern is clear. S = {a,b,c,d, ,z } Roster Method Set of all vowels in the English alphabet: V = {a,e,i,o,u} Set of all odd positive integers less than 10: O = {1,3,5,7,9} Set of all positive integers . | Basic Structures: Sets, Functions, Sequences, Sums, and Matrices Chapter 2 With Question/Answer Animations Copyright © McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Computability Sequences and Summations Types of Sequences Summation Formulae Set Cardinality Countable Sets Matrices Matrix Arithmetic Sets Section Section Summary Definition of sets Describing Sets Roster Method Set-Builder Notation Some Important Sets in Mathematics Empty Set and Universal Set Subsets and Set Equality Cardinality of Sets Tuples Cartesian Product Introduction Sets are one of the basic building blocks for the types of objects considered in discrete mathematics. Important for counting. Programming languages have set operations. Set theory is an important branch of mathematics. Many different
