Copyright © 2014 Curt Hill Sets Introduction to Set Theory.

Copyright © 2014 Curt Hill

Sets

Introduction to Set Theory

Introduction• Fundamental discrete structure• A set is a collection of distinct items• A set has no order• No duplications• An item is in the set or not

– Just as a proposition has two truth values

• Set variables are usually denoted by capital letters and the items by lower case


Terminolgy• Set:

– A collection of distinct items. – Set variable is usually a capital letter. – Braces contain the elements

• Element: aka member– One of the items in a set. – Usually denoted by lower case letters. – Symbol xA, x is member of A.


Terminology 2• Empty set:

– A set with zero members– Symbol is or { }

• Disjoint set: – Two sets with no members in common

• Cardinality:– The number of elements in a set

• Universe of discourse: – The set of all those elements that under

consideration– Often the integers or real numbers.


Subset: • A set whose members are all

contained in another set• The empty set is the subset of every

set• Opposite of superset • A proper subset has at least one

element that is present in the superset and not present in the subset

• An improper subset is the set itself


Notation• A is a (proper) subset of B (AB) • A is a (proper or improper) subset of

B (AB)• A is proper superset of B (A B)• A is superset of B (A B)


Defining a Set• There are typically three ways to

define a set• Enumeration• Set builder• Construction using operators


Enumeration• Lists each element in the set

– A={1,2,3,4,5}

• AKA Roster method• May use an ellipsis to show a large or

infinite set– A={2,4,6,8,…}– A={2,4,6,8,…98,100}


Set Builder Notation• Uses a rule that defines the members

that are present in the set– {x|xI and x>0 and x<5} or

{x|xI and 0 < x <5}– The | is read such that– I is the set of integers

• The expression to the right should give a Boolean value as to whether this is a member or not


Open and Closed • If the type of number is left out, reals

should be assumed– {x|0 < x < 5}

• We cannot say which is the highest and lowest element of this set– We term of this is open– Interval notation is (0,5)

• However the following is closed– {x|0 x 5}– Interval notation is [0,5]


Construction• The third way is to define a set in

terms of others sets using set operations, eg union, intersection, etc

• We will see this as we investigate the operators– Section 1.2 and a different presentation


Power Sets• A power set is the set of all subsets

– Useful for testing all combinations of subsets

• Consider A = { 1, 2, 3}• The power set would then be:

P(A) = {, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }


Tuples• The order of sets is irrelevant

– {1, 2, 3} = {3, 1, 2} = {2, 1, 3}

• In many cases we create ordered tuples• For example, we use Cartesian

Coordinates to indicate a point in two space

• Here order is important– (2,3) is not the same point as (3,2)– This is an ordered pair

• Three space would use an ordered triple


Cartesian Product• A Cartesian Product creates a set of

ordered pairs• Denoted by A ⨯ B• The resulting set of ordered pairs has

all possible combinations where the first element is from A and the second from B


Example• Suppose:

– A = {1, 2, 3}– B = {x, y}

• Then A ⨯ B = { {1,x}, {1,y}, {2,x}, {2,y}, {3,x}, {3,y} }

• Notice that the two sets do not need the same type of elements

• This can be extended to create n-tuples of any size


Connections• In the previous chapter we used the

“is an element of” symbol to show the domain of quantified expressions x(P(x)xx>0)

• This is re-introduced in 2.1 with an addition x(x>0) (P(x))

• The first part restricts the domain to integers greater than zero


Truth Sets• Rosen defines a truth set in a way

similar to a solution set from the Algebra of Real Numbers

• More formally:– Given a predicate P and a domain D– The truth set of P is the set of elements

from D that makes P to be true


Exercises• From 2.1

– 3, 9,19, 23,27,43


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