Sets, Functions and Groups

Set:
A set is generally described as a well-defined collection of distinct objects. By a well-defined collection is meant a collection, which is such that, given any objects, we may be able decide whether the object belongs to the collection or not. By distinct objects we mean objects no two of which are identical.
The objects in a set are called its members or elements. Capital letters A,B,C,X,Y,Z etc., are generally used as names of sets and small letters a,b,c,x,y,z etc., are used as members of sets.

Three Different Ways Of Describing a Set:
1. The Descriptive Method: A set may be described in words. For instance, the set of all vowels of the English alphabets.
2. The Tabular Method: A set may be described by listing its elements within brackets. If A is the set mentioned above, then we may write: A = {a,e,i,o,u}.
3. Set-Builder Method: It is sometimes more convenient or useful to employ the method of set-builder notation in specifying sets. This is done by using a symbol or letter for an arbitrary member of the set and stating the property common to all the members. Thus the above set may be written as: 

A = {x|x is a vowel of the English alphabet}.

This is read as A is the set of all x such that x is a vowel of the English alphabet.

Important Sets:
N= The set of all natural numbers = {1,2,3,...}
W= The set of all whole numbers = {0,1,2,...}
Z= The set of all integers = {0,±1,±2,...} 
Z'= The set of all negative integers = {-1,-2,-3,...}
O= The set of all odd integers = {±1,±3,±5,...}
E= The set of all even integer = {0,±2,±4,...}
Q= The set of all rational numbers = {x|x=p/q where p,q ∈ Z & q ≠ 0 }
Q'= The set of all irrational numbers = {x|x≠p/q where p,q ∈ Z & q ≠ 0 }
ℝ= The set of all real numbers = Q U Q'

Equal Sets:
Two sets A and B  are equal i.e., A=B, if and only if they have the same elements that is, if and only if every element of each set is an element of the other set. Thus the sets {1,2,3} and {2,1,3} are equal.

Important Note:
The phrase if and only if is shortly written as "iff".

Equivalent Sets:
If the elements of two sets A and B can be paired in such a way that each element of A is paired with one and only one element of B and vice versa, then such a pairing is called a one-to-one correspondence between A and B. e.g., 

A = {Ali, Ahsan, Ahmad}

B = {Fatima, Uzma, Saima}

Two sets are said to be equivalent if a (1 - 1) correspondence can be established between them. In the above example A and B are equivalent sets.

Important Note:

Sometimes, the symbol ~ is used to mean is equivalent to. Thus N ~ O.

Singleton Set:

A set having only one element is called a singleton set.

Empty Set or Null set:

A set with no elements (zero number of elements) is called the empty set or null set. The empty set is denoted by the symbol ∅ or {}.

Finite and Infinite Sets:
If a set is equivalent to the set {1,2,3,....,n} for some fixed natural number n, then the set is said to be finite otherwise infinite.
{1,3,5,....,9999} is a finite set.
{1,3,5,...}, sets of number N, Z, Z' etc., mentioned earlier are infinite sets.
Subset:
If every element of a set A is an element of set B, then A is a subset of B. Symbolically this is written as : A ⊆ B (A is subset of B).
In such a case we say B is a super set of A. Symbolically this is written as: B ⊇ A (B is a superset of A).
Proper Subset:
If A is a subset of B  and B contains at least one element which is not an element Of A, then A is said to be a proper subset of B. In such a case we write:  A ⊂ B (A is a proper subset of B).
Improper Subset:
If A is subset of B and A = B, then we say that A is an improper subset of B. From this definition it also follows that every set A is an improper subset of itself.
Important Note:
When we do not want to distinguish between proper and improper subsets, we may use the symbol ⊆ for the relationship.
It is easy to see that: N ⊂ Z ⊂ Q ⊂ R.
Important Statement:
The empty set is a subset of every set.
Power Set:
A set may contain elements, which are sets themselves. For example if: C = Set of classes of a certain school, then elements of C are sets themselves because each class is a set of students. An important set of sets is the power set of a given set.
The power set of a set S denoted by P(S) is the set containing all the possible subsets of S.
Important Note:
The power set of the empty set is not empty.
Universal Set:
When we are studying any branch of mathematics the sets with which we have to deal, are generally subsets of a bigger set. Such set is called the Universal Set or the Universe of Discourse. For example., the set of the rational numbers can be stated as the Universe Set.

Operations On Sets:
Union Of Two Sets: The Union of two sets A and B, denoted by A U B, is the set of all elements, which belong to A or B. Symbolically;

A U B = {x|x ∈ A ∨ x ∈ B}

Thus if A = {1 , 2 , 3} , B = {2 , 3 , 4 , 5} , then A U B = {1 , 2 , 3 , 4 , 5}
Important Note:
Notice that the elements common to A and B, namely the elements 2 , 3 have been written only once in A U B because repetition of an element of a set is not allowed to keep the elements distinct.
Intersection Of Two Sets: The intersection of two sets A and B, denoted by A ∩ B, is the set of all elements, which belong to both A and B. Symbolically;

A ∩ B = {x|x ∈ A ∧ x ∈ B}

Thus if A = {1 , 2 , 3} , B = {2 , 3 , 4 , 5} , then A ∩ B = {2 , 3}

Functions:
A very important special type of relation is a function defined as below:
Let A and B be two non-empty sets such that:

• f is a relation from A to B that is, f is a subset of A x B
• Dom f = A
• First element of no two pairs of f are equal, then f is said to be a function from A to B.

The function f is also written as:

f : A → B

which is read as "f is a function from A to B".

Types Of Functions:

1. Into Function
2. Onto (Surjective) Function
3. (1 - 1) and into (Injective) Function
4. (1 - 1) and Onto Function (Bijective Function)

Set - Builder Notation For a Function:

Consider for instance, the function

f = {(1 , 1), (2 , 4), (3 , 9), (4 , 16),.......}

Dom f = {1, 2, 3, 4,......} and Ran f = {1, 4, 9, 16,......}

This function may be written as: f = {(x , y)|y = x², x ∈ N}

Group:

A monoid having inverse of each of its elements under * is called a group under *. That is a group under * is a set G (say) if

1. G is closed w.r.t. some operation *;
2. The operation of * is associative;
3. G has an identity element w.r.t. * and
4. Every element of G has an inverse in G w.r.t. *.
5. If G satisfies the additional condition: For every a, b ∈ G    a * b = b * a    then G is said to be an Abelian or commutative group under *.