SETS, RELATIONS AND FUNCTIONS

SET : 

A set is a well defined collection of objects. The objects which belong to a set are called members of a set. There are two ways by which the sets can be described.

a) Roster Form - Here the elements of a set are actually written down, separated by commas and enclosed within braces.

eg.

V = {a,e,i,o,u} the set of vowels.

b) Set Builder Form - A set is described by a characterizing property P(x) of its elements x. In such a case, the set is described by {x: P(x) holds}.

eg

A={x: x is an odd number}

NULL SET :

A set consisting of no element at all, is called an empty set or a null set or a void set. It is denoted by ɸ. In roster form, it is written as { }.

FINITE SET :

A is called a finite set, if its elements can be listed by natural numbers 1,2,3 ... and the process of listing terminates in a certain natural number n.

INFINITE SET :

A set whose elements cannot be listed by the natural numbers 1,2,3...n for any natural number n is called an infinite set.

EQUAL SETS :

Two sets A and B are said to be equal, if every element of A is in B  and every element of B is in A. It is written as A=B.

SUBSET :

If A and B are two sets given in such a way that every element of A in B, then A is a subset of set B and it is written as A ⊂ B If at least one element of A does not belong to B, then A is not a subset of B. It is written as A ⊄ B.

SUPERSET :

If a is a subset of B, then we say that B is a superset of A and is written as B ⊃ A.

POWERSET :

The collection of all subsets of a set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.

UNIVERSAL SET :

if there are some sets under consideration then there happens to be a set which is a superset of each one of the given set. Such a set is called the universal set and it is denoted by U.

  

SET OPERATIONS :

i) Union

ii) Intersection

iii) Complement

eg.

Let U (the universal set)  = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}   (a subset of the positive integers)
  A = {2,4,6,8}
  B = {1,2,3,4,5}
  A U B = {1,2,3,4,5,6,8} Union - ALL elements in BOTH sets.
  A ∩ B = {2,4} Intersection - elements where sets overlap.

  A' = {1,3,5,7,9,10} Complement - elements NOT in the set.

  B' = {6,7,8,9,10}

CARDINAL NUMBER :

n, the number of elements in a set is called cardinal number of the set, denoted by n(A).

n(AUB)= n(A)+n(B)-n(A∩B)

n(A-B) = n(A)-n(A∩B)

n(AUBUC) = n(A)+n(B)+n(C)-n(A∩B)-n(B∩C)-n(A∩C)+n(A∩B∩C)   

ORDERED PAIR :

Two elements a and b, listed in specific order, form an ordered pair,described by (a,b).

CARTESIAN PRODUCT :

If A and B are two non-empty sets, then the set of all ordered pairs (a,b) such that a∈A and b∈B is called the cartesian product of A and B. It is denoted by AxB.

RELATION :

The simplest definition of a binary relation is a set of ordered pairs. More formally, a set is a relation if for some x,y. We can simplify the notation and write or simply .
We give a few useful definitions of sets used when speaking of relations.

• The domain of a relation R is defined as , or all sets that are the initial member of an ordered pair contained in R.

• The range of a relation R is defined as , or all sets that are the final member of an ordered pair contained in R.
• The image of a set A under a relation R is defined as .

 

FUNCTIONS :

 

A relation f from A to B ie, subset of AxB, is called a function from A to B, if 

 

 i) for each a∈A there exists b∈B such that (a,b)
∈f

 ii) (a,b)∈f and (a,c)∈f iff b=c

If and is a function, then we can denote this by writing . The set is known as the domain and the set is known as the codomain.

   
CLASSIFICATION OF RELATIONS :

• reflexive: for all x in X it holds that xRx. For example, "greater than or equal to" (≥) is a reflexive relation but "greater than" (>) is not.
• symmetric: for all x and y in X it holds that if xRy then yRx. "Is a blood relative of" is a symmetric relation, because x is a blood relative of y if and only if y is a blood relative of x.
• transitive: for all x, y and z in X it holds that if xRy and yRz then xRz. For example, "is ancestor of" is transitive, while "is parent of" is not. A transitive relation is irreflexive if and only if it is asymmetric.

 

TYPES OF FUNCTIONS :

 

A function f from A to B is called onto if for all b in B there is an a in A such that f (a) = b.   All elements in B are used.

 

 A function f from A to B is called one-to-one (or 1-1) if whenever
 f (a) = f (b) then a = b.   No element of B is the image of more than one element in A.  

 

COMPOSITION OF FUNCTIONS :

 

composite function is denoted g ∘ f : X → Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.  

 

BINARY OPERATIONS : 

 

More precisely, a binary operation on a set S is a map which sends elements of the Cartesian product S × S to S:

Because the result of performing the operation on a pair of elements of S is again an element of S, the operation is called a closed binary operation on S.

Typical examples of binary operations are the addition (+) and multiplication (×) of numbers and matrices as well as composition of functions on a single set. For instance,

• On the set of real numbers R, f(a, b) = a + b is a binary operation since the sum of two real numbers is a real number.
• On the set of natural numbers N, f(a, b) = a + b is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.