ADVANCED TOPICS IN COLLEGE ALGEBRA

 

SET THEORY

SET - A collection of object called ELEMENTS.

∈- indicates membership of an objects in particular set.

   - ( " is an element of " )

∉- indicates non membership of an objects in a particular set.

   - ("is not an element of")

SET NOTATION - ways of describing a set.

1. Tabular/ Roster method : e.g. A = { 1, 2, 3, 4 }

2. Set Builder Notation/ Rule Method : e.g. A = { x I x is a ...}

TYPES OF SETS (10)

1. Finite set - the elements of a set are limited; e.g. A = { a, b, c }
2. Infinite set - the elements of a set are unlimited; e.g. A = { 1, 2, 3, 4 . . . } 
3. Empty/ Null set - there are no element/s in a set; e.g. A = { }
4. Unit set - subset of the universal set.
5. Universal set ( U ) - " the biggest set " 
6. Subset ( ⊇ ) - an aggregate or part of a universal set.

             Concepts:

                    a) A given set is a a subset of itself.
                    b) An empty set is a subset of all sets.
                           e.g. A={ 1, 2, 3 } = 3^2 = 8 subsets.

             * Cardinality of a set - the number of elements of a set.
                           e.g. A={ 1, 2, 3, 4, 5 } . : nA = 5

      7. Equal set - all elements in a set is also the same in the other set;
                   e.g. A = { 1, 2, 3, 4 }    B = { 2, 3, 1, 4 } . : A = B
      8. Disjoint set - No element is the same.
               e.g. A = { 1, 2, 3 }     B = { 4, 5, 6 } . : A ≠ B
     9. Joint set - at least one element of a set is the same w/ the other set.
               e.g. A = { 1, 2, 3, 4 } B = { 5, 6, 7, 1 }
    10. Equivalent set - the number of objects in a set is equal w/ the number of objects 
                                 of the other set. 
               e.g. A = { a, s ,d , f } B = { g, k, l, s }

  SET OPERATIONS 

     1. Union ( U ) - the set that contains everything from any of the sets. 
               e.g. A = { 1, 2, 3, 4 } B = { 5, 6, 7, 8 } . : A U B = { 1, 2, 3, 4, 5, 6, 7, 8 } 

     2. Intersection ( ∩ ) - the set that contains the elements of a set that also belong to
                                    the other set.
               e.g. A = { a, b, c } B = { b, c, d } . : A ∩ B = { b, c }

     3. Relative complement ( - ) - the theoretic difference of a set to the other set.
               e.g. A = { 1, 2, 3, 4 } B = { 3, 4, 5, 6 }

                 . : A-B = { 1, 2 }    . : B-A = { 5, 6 }

     4. Complement of a set ( ' ) - a set X that contains the elements of the universal 
                                                set( U ) which is not the same to its original elements.
               e.g. U = { a, b, c, d, e, f }     A = { g, h, k, j, c, d, e }

                      .: A' = { a, b, f }

     5. Set product ( cross product )
               e.g. A = { a, b } B = { x, y, z }

               .: A x B = { ( a, x ), ( a, y ), ( a, z ), ( b, x ), ( b, y ), ( b, z) }

VENN DIAGRAM

                  - It is a very useful pictorials in dealing w/ 

                    operations involving sets.

                 - After the Cambridge logician John Venn

                 (1834 - 1923)

 

         UNIVERSAL SET - represented by a rectangle.

           SUBSET - drawn as a circles w/in the rectangle.