Sets and Set Operations

Section 2.1

 

Set:  a collection of objects

Elements:  members belonging in a set

Sets can be well-defined (without ambiguity) or not well-defined.

 

Notations: 

        S = {a, b, c }  

represents a list, roster.

The set S is equal to the set of elements: a, b, c.

 

Set builder notation:

G     =     {x | x > 0 }

reads:  "The set G is equal to the set of x values such that x is greater than 0."

So G is the set of all positive numbers, which is impossible to list, therefore set builder notation is necessary.

 means is an element of (belongs in)

 means is not an element of (does not belong in)

 

The cardinal number of a set is the number of elements in the set.   What is the cardinal number of set S above?          

  n(S) = 3.

What is the cardinal number of set G above?

        n(G) =

c  {a, b, c}                c  S

f   { a, b, c}               f   S

T = { c, b, a}

T = S  

 

Two sets are equal if and only if every element in one set is in the other.

So every element in T must be in S and every element in S is found in T.

Empty Set: Set containing no elements is designated , the null set.

        n(  ) = 0

If 

        E = {   }         

Then 

        n(E) = 0

Note: the set Z = { 0 }  is not an empty set.  n(Z) = 1, there is one element in the set Z and it is the element 0.

Universal Sets: U is the set of all possible elements used in the problem.

Example 1: 

Let 

        U = { x | x is a student in this class }

Let 

        A = { x | x is a student in the first row }

        B = { x | x is a student majoring in Liberal Arts }

        C = { x | x is a student over 7 feet tall }

 

A Subset is a set that is contained in the Universal set.

A, B, C are all subsets of U.

A  U,         B  U,         C  U

And in general        U,     or any set.

Venn Diagrams:

 

 

 

 

 

 

 

Operations with sets.

Intersection and Union:

The Intersection of two sets is the overlap of the sets; what they both have in common. 

        A B = { x | x  A and x  B }

The Union of two sets consists of all elements in A or B or both.

 

        A B = { x | x  A or x  B }

 

Mutually exclusive sets have nothing in common. 

If 

        W = { 1, 2, 3 } 

and 

        S = { a, b, c }

Then 

        W  S =

 

The cardinal number of a union of sets:

        n(A B ) = n(A) + n(B) n(A B)        

otherwise some elements are counted twice.

 

The Complement of set A: 

Designated A (A prime),

The elements that are in the Universal set, but not in the specific set, A.

 

Example 2: 

Let 

        U = { x |  x is a letter in the alphabet }

        A = { x | x is a consonant }

        A = { x | x is not a consonant }

 

Cardinal Number of Complements

        n(A) + n(A) = n(U)

So 

        n(A) = n(U) n(A)

Find n(A).  

        n(U) n(A) = n(A)        

        26   5  =  21

 


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