SET THEORY

Definition:
A set is a collection of objects or members, which are related in some
way. Sets are denoted by capital letters, e.g. set A, B, M etc.
Terms used:

  1. Member (element) of a set
    The objects in a set are called members or elements of the set.
    Members of a set are enclosed in curly brackets. The symbol Є is a
    short form of saying ‘’is a member of’’ and is for ‘’not a member
    of’’. Consider set A = {1, 3, 4, 6}. Then 1ЄA, 3ЄA, 4ЄA, and 6Є A.
    Whereas 7, 8, 9 etc are not members of set A, i.e. 7 A, 8 A, 9 A,
    etc.
  2. Subsets
    Set A is said to be a subset of set B if every element of set A is also in
    set B. E.g. given that B = {1, 2, 3, 4, 5, 6, 7, 8} and A = {1, 3, 5, 7}. Here
    every element of set A is also in set B. Therefore, set A is a subset of
    B. The symbol < or > is for ‘’subset of’’ and is for ‘’not a subset
    of’’. Therefore A < B or B > Abut B A because not all elements of
    B are in A.
  3. Empty set (null set)
    An empty set is a set with no element. It is at time called null set. The
    null set is denoted by the symbol { } or ф.
    Note:
    The empty set { } is not the same as {0}. This is because the set {0} has
    one element which is 0 whereas the set { } has no element.
  4. Finite sets
    The set is called finite if the elements of the set can be counted.
    Example
    Consider the following sets:
    D = {days of the week}
    F = {factors of 12}
    G = {whole numbers greater than 5 but less than 11}
    We can list all the members of these sets.
    D = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday,
    and Sunday}

F = {1, 2, 3, 4, 6, and 12}
G = {6, 7, 8, 9, and 10}

  1. Infinite sets
    These are sets with unlimited number of elements.
    Example
    Given the following sets:
    W = {whole numbers}
    R = {real numbers}
    M = {multiple of 3}
    Here, we cannot list all the members of these sets.
    W = {0, 1, 2, 3, 4, 5, 6 ……}
    R = {……-2, -1, 0, 1, 2 …}
    M = {3, 6, 9, 12 ……}
    All members of these sets cannot be exhausted so they are infinite
    sets.
  2. Number of elements in a set
    The number of element in a finite set can be counted. The number of
    elements of set A is denoted by n A and it is the total number of
    elements in set A.
    Example
    Find the number of elements in the following sets.
    a) R = {1, 2, 3, 4, 5, 6, 7, 8, 12}
    b) B = {2, 4, 6, 8, 9}
    Solution
    a) n (A) = 9
    b) n (A) = 5
    Example
    Given that set B = {factors of 24}
    a) Write out set B in full
    b) Find n (B)
    Solution
    a) B = {1, 2, 3, 4, 6, 8, 12, 24}
    b) n (B) = 8
    Example
    Given that set N = {natural numbers from 2 to 11}

a) Write out set N in full
b) Find n (N)
Solution
a) N = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}
b) n (N) = 10

  1. Equal sets
    Two or more sets are equal if they contain the same elements.
    E.g. A = [1, 3, 5, 7} and B = {1, 3, 5, 7} are equal sets. Here A = B and
    also B = A
  2. Equivalent sets
    Two or more sets are said to be equivalent if they contain the same
    number of elements. E.g. set A = {a, e, i, o, u} and B = {2, 4, 6, 8, 10}.
    Sets A and B contain the same number of elements which is 5. We
    therefore say that they are equivalent sets.
  3. Union of sets ( u )
    The union of two sets is the set of all elements that are members of
    either set. The symbol for union is U.
    Example
    Given that: M = {1, 2, 3, 4} and N = [3, 4, 6, 7}
    i) List M U N
    ii) Find n M U N
    Solution
    i) M U N = (1,2,3,4,6,7)
    ii) n (M U N) = 6
  4. Intersection of sets ( n )
    The intersection of two sets or more sets is the set of elements that
    are in both sets.
    Example
    Given two sets: A = {–1, 0, 4, 5, 6, 7} and B = {–1, 6, 8, 10}
    Find:
    i) n( A u B)
    ii) n(A n B)
    Solution
    i) (A u B) = { -1,0,4,5,6,7,8,10} n(A u B) = 8
    ii) (A n B) = {1,6} .: n (A n B) = 2
  5. Disjoint set
    When the intersection of the two sets is empty, the two sets are
    called disjoint sets. E.g. given that P = {1, 3, 5, 7} and Q = {2, 4, 6, 8}.
    Here( P n Q) = { }
  6. Complement of set
    Consider two sets: A = {a, b, c, d} and B = {a, b, c, d, e, f}.
    Members which are present in B and not present in A is called
    complement of A denoted by Á or Â. From the two sets above:
    A’ = { e,f } therefor n(A’ ) = 2
    Also:
    A’ n B = {e ,f } = n (A’ n B ) = 2
  7. The universal set (ℇ)
    This is a set that contains all the members of an item or object under
    consideration. It is denoted by the symbol ℇ.