Set Theory

SET

A set is a well-defined collection of objects or elements. Each element in a set is unique. Usually but not necessarily a set is denoted by a capital letter e.g., A, B, ....., U, V etc. and the elements are enclosed between brackets { }, denoted by small letters a, b, ....., x, y etc. For example:

A      =      Set of all small English alphabets
=      {a, b, c, ....., x, y, z}
B =      Set of all positive integers less than or equal to 10
        =      {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
R      =      Set of real numbers
        =      {x : - ∞ < x < ∞}

The elements of a set can be discrete (e.g. set of all English alphabets) or continuous (e.g. set of real numbers). The set may contain finite or infinite number of elements. A set may contain no elements and such a set is called Void set or Null set or empty set and is denoted by Φ(phi). The number of elements of a set A is denoted as n(A) and hence n(Φ) = 0 as it contains no element.

Union of Sets

Union of two or more sets is the best of all elements that belong to any of these sets. The symbol used for union of sets is 'U'

i.e. AUB = Union of set A and set B
= {x : x ε A or x ε B (or both)}

e.g. If A = {1, 2, 3, 4} and B = {2, 4, 5, 6} and C = {1, 2, 6, 8} then AUBUC = {1, 2, 3, 4, 5, 6 8}.

Intersection of Sets

It is the set of all of the elements, which are common to all the sets. The symbol used for intersection of sets is '∩'.

i.e. A∩B = {x : x ε A and x ε B}
e.g. If A = {1, 2, 3, 4} an B = {2, 4, 5, 6} and C = {1, 2, 6, 8}, then A∩B∩C = {2}.
Remember that n(AUB) = n(A) + n(B) -(A∩B).

Difference of Two Sets

The difference of set A to B denoted as A - B is the set of those elements that are in the set A but not in the set B i.e. A - B = {x : x ε A and x

A}.
In general A-B≠B-A
e.g. If A = {a, b, c, d} and B = {b, c, d} then A - B = {a, d} and B - A = {e, f}.

Subset of a Set

A set A is said to be a subset of the set B is each element of the set A is also the element of the set B. The symbol used is '

' i.e. A

B <=> (x ε A => x ε B).

Each set is a subset of its own set. Also a void set is a subset of any set. If there is at least one element in B which does not belong to the set A, then A is a proper subset of set B and is denoted as A

B.

e.g. If A = {a, b, c, d} and B = {b, c, d} then B A or equivalently A

B (i.e. A is a super set of B).

Equality of Two Sets

Sets A and B are said to be equal if A

B and B

A and we write A = B.

Universal Set

As the name implies, it is a set with collection of all the elements and is usually denoted by U. e.g. set of real numbers R is a universal set whereas a set A = [x : x < 3} is not a universal set as it does not contain the set of real numbers x > 3. Once the universal set is known, one can define the Complementary set of a set as the set of all the elements of the universal set which do not belong to that set. e.g. If A = {x : x < 3 then

(or A^c) = complimentary set of A = {x : x > 3}. Hence we can say that A U

= U i.e. Union of a set and its complimentary is always the Universal set and A ∩

= f i.e. intersection of the set and its complimentary is always a void set. Some of the useful properties of operation on sets are as follows:

operation-on-sets