Introduction to Set Theory

Introduction to Sets – the basics

A set is a collection of objects.

Objects in the collection are called elements of the set.

Examples - setThe collection of persons living in Pembroke Pines is a set.

◦Each person living in Pembroke Pines is an element of the set.

The collection of all counties in the state of Florida is a set.◦Each county in Florida is an element of the set.

Examples - setThe collection of counting numbers is a set.

◦Each counting number is an element of the set.

The collection of pencils in your backpack is a set.◦Each pencil in your backpack is an element of the set.

Notation Sets are usually designated with capital letters.

◦ A = {a, e, i, o, u}

Elements of a set are usually designated with lower case letters.◦ In the set A above, the individual elements of the set are designate as, a, e, i, etc.

Roster MethodFor example, the set of counting numbers from 1 to 5 would be written as {1, 2, 3, 4, 5}.

For example, the set of animals found on a farm would be written as {chickens, cows, horses, goats, pigs }.

Notation Set builder notation has the general form {variable | descriptive statement }.

The vertical bar (in set builder notation) is always read as “such that”. Set builder notation is frequently used when the roster method is either inappropriate or inadequate.

Example – set builder notation {x | x < 6 and x is a counting number} is the set of all counting numbers less than 6. Note this is the same set as {1,2,3,4,5}.

Notation – is an element of If x is an element of the set A, we write this as x A. x A means x is not an element of A. If A = {3, 17, 2 } then 3 A, 17 A, 2 A and 5 A. If A = { x | x is a prime number } then 5 A, and 6 A.

Notation – is a subset of The set A is a subset of the set B if every element of A is an element of B. If A is a subset of B we write A B to designate that relationship. If A is not a subset of B we write A B to designate that relationship.

Example - subset The set A = {1, 2, 3} is a subset of the set B ={1, 2, 3, 4, 5, 6} because each element of A is an element of B. We write A B to designate this relationship between A and B. We could also write {1, 2, 3} {1, 2, 3, 4, 5, 6}

Example – not a subset of The set A = {3, 5, 7} is not a subset of the set B = {1, 4, 5, 7} because 3 is an element of A but is not an element of B.

The empty set is a subset of every set, because every element of the empty set is an element of every other set.

Definition – universal set A universal set is a set which contains all objects under consideration, including itself, and is denoted as an uppercase U. For example, if we define U as the set of all living things on planet earth, the set of all dogs is a subset of U.

Venn DiagramsIt is frequently very helpful to depict a set in the abstract as the points inside a circle (or any other closed shape).

We can picture the set A as the points inside the circle shown here.

A

Definition – universal set A universal set is a set which contains all objects, including itself. In a Venn Diagram, the Universal Set is depicted by putting a U box around the whole thing.

Operations on sets There are three basic set operations: - Intersection

- Union- Complement

Definition – intersection of The intersection of two sets A and B is the set containing those elements which are elements of A and elements of B.

We write A B

Example - intersectionIf A = {3, 4, 6, 8} and B = { 1, 2, 3, 5, 6} then A B = {3, 6}

Venn Diagram - intersectionA is represented by the red circle and B isrepresented by the blue circle. When B is moved to overlap a A portion of A, the purple colored region B illustrates the intersection of A and B A ∩ B

Definition – union of The union of two sets A and B is the set containing those elements which are elements of A or elements of B.

We write A B

Example - unionIf A = {3, 4, 6} and B = { 1, 2, 3, 5, 6} , then A B = {1, 2, 3, 4, 5, 6}.

Venn Diagram - unionA is represented by the red circle and B is represented by the blue circle. AThe purple colored region illustrates the intersection. B

The union consists of allpoints which are colored red or blue or purple. A B

Definition – complement of A complement of a set is the set of those elements which does not belong to a given set but belongs to the universal set. A complement is denoted by A’, A, or ~ A. The number of elements of A and the number of elements of A’ make up the total number of elements in U.

Venn Diagram – A’

•Given the following Venn diagram, shade in (B UC) ~ A.

First work with what is inside the parentheses. The union of B and C shades both circles fully.

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Then do the complement A by cutting out the overlap with A.

Definition – cross product The cross product of two sets A and B, denoted by A x B (read A cross B), is defined as the set of all ordered pairs (a,b) where a is a member of A and b is a member of B.

For example: A = {1, 2, 3} B = {a,b} A x B = {(1,a), (1,b), (2,a), (2,b), (3,a), (3,b)}

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Algebraic Properties Union and intersection are associative operations.

(A B) C = A (B C)

(A ∩ B) ∩ C = A ∩ (B ∩ C)

Algebraic Properties Union and intersection are commutative operations.

A B = B A A ∩ B = B ∩ A

Algebraic Properties Two distributive laws are true.

A ∩ ( B C )= (A ∩ B) (A ∩ C) A ( B ∩ C )= (A B) ∩ (A C)