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Set Theory: Basic Concepts Reviews

Symbols and Terminology

A set is a group that contains objects called the elements.  A set of odd counting numbers is expressed as {1, 3, 5, 7, 9} or by the notation {x│x is an odd counting number less than 10}.

                                                                        ↑

                                                                     where

*Make sure you understand the definition of counting numbers, whole numbers, integers, rational numbers, real numbers, and irrational numbers. Check your text for definitions. 

 

Venn diagrams and subsets

Venn diagrams are used to illustrate various sets and their relationships.  They contain a universal set U, and usually have particular set(s) within U that are represented by circle or oval regions.  

Concept 1: The Complement of a Set

For any set A within the universal set U, the complement of A, written as A' (pronounced as A prime), is the set of elements of U that are NOT elements of A.  It can be expressed as

                        A' = {x│x is an element of U and x is not an element of A}

 

For example:

            U = counting number less than 10 = {1,2,3,4,5,6,7,8,9}

            A = prime number = {2,3,5,7}

            A' = NOT prime number = {1,4,6,8,9}

 

 

 

Concept 2: Subset and Proper Subset

Set A is a subset of set B if every element of A is also an element of B.

Set A is a proper subset of B if every element of A is also an element of B, but A CANNOT be exactly the same as B.

 

For example:

            A = {a,b,c,d,e}

            B = {a,b,c,d,e,f}

A is said to be a subset AND a proper subset of B.

NOTE: The number of subsets of a set with n elements is 2n.

For example: If a set has 5 elements, it will have 25 or 32 subsets.

The number of proper subsets of a set with n elements is 2n - 1.

 For example: If a set has 5 elements, it will have 25 - 1 or 31 proper subsets.

 

Set Operations And Cartesian Products

Concept 3: Intersection of Sets

The intersection of sets A and B, denoted as AB, is the set of elements common to both A AND B.

 

For example:

            A = {1,2,3,4,5}

            B = {2,4,6,8,10}

The intersection of A and B (i.e. AB) is simply {2, 4}.

 

Concept 4: Union of Sets

The union of sets A and B, written as AUB, is the set of elements that appear in either A OR B.

For example:

            A = {1,2,3,4,5}

            B = {2,4,6,8,10}

The union of A and B (i.e. AUB) is {1, 2, 3, 4, 5, 6, 8, 10}.

 

 Concept 5: Difference of Sets

The difference of sets A and B, written as A-B, is the set of elements belonging to set A and NOT to set B.

For example:

            A = {1,2,3,4,5}

            B = {2,3,5}

The difference of A and B (i.e. A-B) is {1,4}.

NOTE: A-B ≠ B-A

 

Concept 6: Cartesian Product of Sets

The Cartesian product of sets A and B, written A x B, is expressed as:

A x B = {(a,b)│a is every element in A, b is every element in B}

For example:

            A = {1,2}

            B = {4,5,6}

The Cartesian product of A and B (i.e. A x B) is {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)}.

 

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