Set Theory

Introduction: The notion of a set is elementary to all of Mathematics and every branch of mathematics can be considered as a study of sets of objects of one kind or another. Cantor was the founder of the theory of sets. The word set is a primitive term and is regarded as one of the basic undefined ideas of mathematics. But we must have an Intuitive idea of what we mean by a set. Let us now consider the idea of a set.

Sets and Operations on Sets

Definition: A set is collection of well defined objects. In the above definition the words set and collection for all practical purposes are Synonymous. We have really used the word set to define itself.

Notations: Each of the objects in the set is called a member of an element of the set. The objects themselves can be almost anything. Books, cities, numbers, animals, flowers, etc. Elements of a set are usually denoted by lower-case letters. While sets are denoted by capital letters of English larguage.
The symbol ∈ indicates the membership in a set.
If “a is an element of the set A”, then we write a ∈ A.
The symbol ∈ is read “is a member of ” or “is an element of ”.
The symbol ∉ is used to indicate that an object is not in the given set.
The symbol ∉ is read “is not a member of ” or “is not an element of ”.
If x is not an element of the set A then we write x ∉ A .

Specifying Sets: There are five different ways of specifying sets:
(i) One method of specifying a set is to list all the members of the set between a pair of braces. Thus {1, 2, 3} represents a set. This method is called “The listing method”.

Example 1:
(i) {3, 6, 9, 12, 15}
(ii) {a, b, c, d}
This method of listing the elements of the set is also known as ‘Tabulation’. In this method the order in which the elements are listed is immaterial, and is used for small sets.
(ii) Another method of defining particular sets is by a description of some attribute or characteristic of the elements of the set. This method is more general and involves a description of the set property.
A = {x | x has the property P}
Designates “the set A of all objects ‘x’ such that x has the property P”. This notation is called Set- Builder notation. The vertical bar | is read as “such that”.

Example 2:
(i) A = {x | x is a positive Integer greater then 100}.
This is read as “the set of all x is a positive Integer less than 25”.
(ii) B = {x | x is a complex number}.
Note: Repetition of objects is not allowed in a set, and a set is collection of objects without ordering.
(iii) We can describe a set by its characteristic function.

(iv) In another method we describe the set by a recursive formula:
Example 3: Let x0 = 2, x1 = 1 and 1 –1; 1 xi = xi xi i ≥ and A = {xi : i ≥ 0}.
(v) We can also describe a set by an operation on some other sets.

Subsets: Definition : A set A is a subset of the set B if and only if every element of A is also an element of B.
We also say that A is contained in B, and use the notation A ⊆ B.
Symbolically: If x ∈ A ⇒ x ∈ B, then A ⊆ B.
If A ⊆ B , it is possible that A = B, to emphasize this fact we write A ⊆ B.
If A is contained in B, then we may also state that B contains A and write B ⊇ A.

Proper Subsets
Definition : A set A is called proper subset of the set B. If (i) A is subset of B and (ii) B is not a subset A
i.e., A is said to be a proper subset of B if every element of A belongs to the set B, but there is atleast one element of B, which is not in A. If A is a proper subset of B, then we denote it by A ⊂ B.
Note: Every set is a subset to itself.

Equal Sets: If A and B are sets such that every element of A is an element of B and every element of B is an element of, A then A and B are equal (Identical). We write “A = B”, and it is read as A and B are identical.
Super sets: If A is subset of B, then B is called a superset of A.
Example:
(i) If A = A{0, 2, 9}, B = {0, 2, 7, 9, 11} then A ⊂ B (A is a proper subset of B).
(ii) If A = {a, a, b}, B = {a, b}, then A and B denoted the same set, i.e., A = B.
(iii) If A = {1, 2, 4}, B = {2, 4, 6, 8} A is proper subset of B and B is a superset of A.

Null Set
Definition: The set with no elements is called an empty set or null set. A Null set is designated by the symbol φ .
The null set is a subset of every set, i.e., If A is any set then φ ⊂ A.
Example:
(i) The set of real roots of the polynomial x2 9 = 0.
(ii) {x | 5x = 5x 2}.