Calculus- Early Transcendentals, 2021a

Chapter 1
Review
Success in calculus depends on your background in algebra, trigonometry, analytic geometry and functions.
In this chapter, we review many of the concepts you will need to know to succeed in this course.
1.1 Algebra
1.1.1 Sets and Number Systems
A set can be thought of as any collection of distinct objects considered as a whole. Typically, sets are represented
using set-builder notation and are surrounded by braces. Recall that (,) are called parentheses
or round brackets; [,] are called square brackets; and{,} are called braces or curly brackets.
Example 1.1: Sets
The collection {a,b,1,2} is a set. It consists of the collection of four distinct objects, namely, a, b,
1 and 2.
Let S be any set. We use the notation x ∈ S to mean that x is an element inside of the set S, andthe
notation x ∉ S to mean that x is not an element of the set S.
Example 1.2: Set Membership
If S = {a,b,c},thena ∈ S but d ∉ S.
The intersection between two sets S and T is denoted by S ∩ T and is the collection of all elements
that belong to both S and T .Theunion between two sets S and T is denoted by S ∪ T and is the collection
of all elements that belong to either S or T (or both).
Example 1.3: Union and Intersection
Let S = {a,b,c} and T = {b,d}. ThenS ∩ T = {b} and S ∪ T = {a,b,c,d}. Note that we do not
write the element b twice in S ∪ T even though b is in both S and T .
Numbers can be classified into sets called number systems.
N the natural numbers {1,2,3,...}
Z the integers {...,−3,−2,−1,0,1,2,3,...}
{ }
Q the rational numbers Ratios of integers: p
q : p,q ∈ Z,q ≠ 0
R the real numbers Can be written using a finite or infinite decimal expansion
C the complex numbers These allow us to solve equations such as x 2 + 1 = 0
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