ELEMENTARY ALGEBRA
MAT 0024
Prof. Howard Sorkin

NOTES ON SETS
EXPLANATION FOR SUPPLEMENTARY SHEET 1

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Lecture for Definitions 1 - 8
(7 min - please allow time for streaming to start)
Lecture for Definitions 9 and 10 - Union and Intersection
  (13 min - please allow time for streaming to start) )


  1.  
SET:   A collection of things.

  2.  ELEMENT: An element is MEMBER of a SET.   It is denoted by the symbol
    If something is NOT AN ELEMENT of a ser we use the symbol   

  3.   The NOTATION for sets are BRACES, {  }

Example:  {Jan., Feb., Mar.}    The set of the first three months of the year.

Example:  {a, e, i, o, u}              The set of the vowels in the English language.

  4.   CAPITAL letters are used to name a set: 

Example:  A  =  {Jan., Feb., Mar.}
                    B  =  {a, e, i, o, u}

  5.   SUBSET:  If every element in a given set B is also an element of a set A, then B is a
                         
  SUBSET of A.

  Example: BA  means "B is a SUBSET of A."

If H  =  {1, 2, 3, 4, 5, 6}
and
K  =  {2, 4, 6}
then K H 
 
  Note 1 If there is at least one element of B that is not in A then B is NOT A SUBSET of A.
This is written BA
     
 
Example:
If H = {1, 2, 3, 4, 5, 6} and
M = {5, 7, 9}

then MH
     
  Note 2  Every set is a subset of itself.

   6.   The NULL or EMPTY SET is a set which contains NO ELEMENTS.
         The symbol for the empty set is f the Greek letter Phi or we may just  write {  }.

   7.   The UNIVERSE or UNIVERSAL SET is the set which contains all the 
        elements under  discussion.  The symbol for the UNIVERSAL SET is

  8.   The COMPLEMENT of a set A is the set of all elements in the UNIVERSAL SET
         which are
NOT in the set A.

        The COMPLEMENT of a set is denoted as A' and can be read as "A complement,"
        "the complement of A," or "A prime."

  Example: If U = {a, e, i, o, u} and
A = {a, i, u}
then A' = {e, o},
since the elements"e" and "o" are the only elements that are NOT in A but yet are in the Universal Set U.  

  9.    The UNION of sets A and B, AB, is the set containing all the 
         elements that are members of set A OR set B. 

10.  The INTERSECTION of two sets A and B,  AB , is the set containing
        all
 the elements that are common to BOTH sets A AND set B. 

Below is an example of UNION and INTERSECTION using the following sets: 

   =  {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 
  A  =  {1, 2, 4, 6} 
  B  =  {1, 3, 6, 7, 9} 
  C =   {  }

    
AB = {1, 2, 3, 4, 6, 7, 9} AB = {1,6} 
AC = {1, 2, 4, 6} = A  AC = {  } = C 
A'B = {1, 3, 5, 6, 7, 8, 9, 10} 
    since A' = {3, 5, 7, 8, 9, 10} 
A'B = {3, 7, 9} 
(AB)' = {5, 8, 10}  (AB)' = {2, 3, 4, 5, 7, 8, 9, 10} 
 
 
You should now be ready to do the exercises on
SUPPLEMENTARY SHEET 1
 

© Copyright 1986, 1998, 2001, 2005, 2006 Howard Sorkin All rights reserved