Relations and Functions | Injective | surjective | Commutative Laws

In this article Relations and Functions we give the information about Relation and Function introduction and different set laws such as Idempotent laws, Identity laws, Commutative Laws, Associative laws. 

Relations and Functions:

Sets Theory operations such as of union, intersection, and complement satisfy various laws (identities) which are listed in below Table

Set Theory Laws

Idempotent Laws

P ∪ P = P

P ∩ P = P

Associative Laws

(P ∪ Q) ∪ R = P ∪ (Q ∪ R)

(P ∩ Q) ∩ R = P ∩ (Q ∩ R)

Commutative Laws

P ∪ Q = Q ∪ P

P ∩ Q = Q ∩ P

Distributive Laws

P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)

P ∩ (Q ∪ R) =(P ∩ Q) ∪ (P ∩ R)

De Morgan’s Laws

(P ∪ Q)c=Pc∩ Qc

(P ∩ Q)c=Pc∪ Qc

Identity Laws

P ∪ ∅ = P
P ∪ U = U

P ∩ U =P
P ∩ ∅ = ∅

Complement Laws

P ∪ Pc= U
P ∩ Pc= ∅

Uc= ∅
c = U

Involution Law

(Pc)c = P

 

Relationship in a set:

Any relation R on any set A is a subset of A X A.

Now let us discuss the relation between two sets –

Let A = {1 , 8 ,10} and B = {1 , 2 , 3} be two sets then

A x B = {(1,1) , (1 ,2) , (1,3) , (8,1) , (8,2) , (8,3) , (10,1) , (10,2) , (10,3)}

Now any relation from A to B is found as (x , y) R ,

where x = y2 {x A , y B} i.e. in ordered pair (x , y)

First Element = (Second Element)2

So R = {(1,1) , (8,2) , (10,3)}

It is clear that R A x B

Definition : Any relation R from set A to set B is a subset of A x B i.e. R A x B

Injective, surjective, Bijective of Functions:

A function f from P to Q is an assignment of exactly one element of Q to each element of P  also must P and Q are non-empty set. P is called Domain of f and Q is called co-domain of f. If q is the unique element of Q assigned by the function f to the element a of P, it is written as f(p) = q. f maps P to Q. means f is a function from P to Q.

Terms related to functions:

1. Domain and co-domain – if f is a function from set P to set Q, then P is called Domain and Q is called co-domain.

2. Range– Range of f is the set of all images of elements of P. Basically Range is subset of co- domain.

3. Image and Pre-Image– q is the image of p and p is the pre-image of q if f(p) = q.

Properties of Function:

  1. Addition and multiplication: let f1 and f2 are two functions from P to Q, then f1 + f2 and f1.f2 are defined as-:
    f1+f2(x) = f1(x) + f2(x). (addition)
    f1f2(x) = f1(x) f2(x). (multiplication)
  2. Equality: Two functions are equal only when they have same domain, same co-domain and same mapping elements from domain to co-domain.

Relations and Functions:

Types of functions:

1. Injective:

One to one function(Injective): A function is called one to one if for all elements p and q in P, if f(p) = f(q),then it must be the case that p = q. It never maps distinct elements of its domain to the same element of its co-domain.

Injective
Injective
  1. We can express that f is one-to-one using quantifiers as or equivalently  , where the universe of discourse is the domain of the function.

2. Surjective:

Onto Function (surjective): If every element q in Q has a corresponding element p in P such that f(p) = q. It is not required that a is unique; The function f may map one or more elements of P to the same element of Q.

surjective
surjective

Bijective/Invertible:

3. One to one correspondence function(Bijective/Invertible):A function is Bijective function if it is both one to one and onto function.

Bijective
Bijective

4. Inverse Functions: Bijection function are also known as invertible function because they have inverse function property. The inverse of bijection f is denoted as f-1. It is a function which assigns to q, a unique element a such that f(p) =q. hence f-1(q) = p.

Related Links :

  1. complement of a set | types of set theory | Roster method | universal set
  2. Operation on Sets | Union of sets | Intersection of sets | Difference of sets
  3. cartesian product of a set | venn diagram math | define cartesian product
  4. Age problems tricks || word problems ||age problems questions
 

 

Santosh Nalawade

Work as Assistant Professor and Web Developer.

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