1.1-1.6, due on September 5

The most difficult part of the material I would say is the indexed collections of sets.  It is a new notation and concept which I was not previously familiar with, but it does give a simpler way to express a union of multiple sets which is very useful

The most interesting is the last part about the Cartesian products of sets, especially how sets can be used to express Cartesian coordinates and also the Euclidean plane.  This related directly to the use of sets to describe multidimensional points and then later vectors which included a magnitude. We used these a lot in linear algebra and multi-variable calculus.  It also explains the notation R^2 which came up a lot to express the Euclidean plane or later R^3.





1.1 Describing a Set

• sets contain a small number of elements
• sets without an element is the null set
• we can define a set by  S={x: p(x)} where the elements must satisfy some condition
• different sets include N natural numbers (positive integers), Z integers, Q rational numbers, I irrational numbers, R real numbers, and C complex numbers

1.2 Subsets

• A is called a subset of a set if every element of a set also belongs to the other set. 
• U is called the universal set
• The interval (a,b) = {x ∈ R: a < x < b}.
• Set of all sets is the power set
• A is a proper set of B if A is a subset but not equal to B
• Venn diagrams help visualize sets

1.3 Set Operations

• A U B is adding the sets together
• A ∩ B  is the intersection of the sets
• If the intersection is the null set then they are disjoint
• A−B = {x: x ∈ A and x ∈ / B}. 
• A complement of a set is the universal set U minus the set

1.4 Indexed Collections of Sets

• When finding the union of multiple sets use the notation/ and also intersection with opposite sign

n Ui=1 Ai ={x : x ∈ Ai for some i, 1 ≤ i ≤ n}. 

• I can be used as an index set.  The following is an indexed collection of sets.

USα = {x : x ∈ Sα for some α ∈ I}, α∈I 

1.5 Partition of Sets

• pairwise disjoint means that every two distinct subsets are also disjoint.
• pairwise disjoint where every element belongs to a subset is called a partition.

1.6 Cartesian Products of Sets

• an ordered pair is the set (x,y)
• A cartesian product is when A × B= { ( a , b ) : a ∈ A and b ∈ B }