Relations and functions

Relations and functions

— Objective —

• Learn about relations that qualify to be functions.
• How to link pairs of objects from 2 sets in a pair.

— Introduction —

• In real life, we come back across many examples of relations –
• Employer and Employee
• Mother & Daughter

• Similarly, in Maths, we’ve relations like

•  Line B

• Functions is also a crucial part in Mathematics since it defines precise correspondence between two quantities.

— Products of sets —

• Let P and Q be two non-empty sets and  , , then P*Q = {(a,c) : (a,d) : (b,c) : (b,d)}

• If P =  or Q = , then P*Q =

–Important Point–

1. Let A = {x,y} and B = {a,b}, then A = B only if x = a  and y =b.

• Let there be a Set A and  then A*A*A  = {(a,b,c)}, also note that (a,b,c) is called ordered triplet.

• If  and either A or B is an infinite set, then A*B = (infinite set).
• Let n(A) = p and n (B) = q, then n (A*B) = pq.

— Relations –

• They are represented in 3 ways – Roster Form, Set Builder Form and Arrow Diagram.

Example 1 – Let there be two set P and Q such that P = {a,b,c} and Q = {Annu, Bharat, Bhanu, Chandra, Divya}.

• Roster Form :- Relation R = {(a,Annu),(a,Bharat),(a,Bhanu) . . . . . . . . . . . . (c,Divya)}

• Set Builder Form :-  R = {(x,y) : x is the first letter of the name,  and
• Arrow Diagram :- Shown Below.

— Definitions related to Relations –

• Domain – The set which comprise of the first elements in relation R is called Domain.
  • In Example 1, the set {(a,b,c)} is the domain
• Range – The set which comprises of the second elements in relation R is called range.
  • In Example 1, the set {(Annu, Bharat, Bhanu, Chandra, Divya)} is the range
• Codomain – Second set is also known as codomain.
  • In Example 1, the set {(Annu, Bharat, Bhanu, Chandra, Divya)} is the codomain

— Important Points related to Relations –

1.   
2.  If n(A) = p and n(B) = q, then n(A*B) = pq , also total number of relations between set A and set B are 2^pq.

— Functions –

• Definition – A relation from X  to Y  such that domain of the function is X and no same previous element is present.

• Let f be a function from X to Y, (x,y) , then f(x) = y,  y is called the image of x under f and x  is known as preimage of y under f

Example of a function :- f:A—->B

• A function f  with range = R (set of real numbers) or one of its subsets is called a real valued function and a function with with range = R (set of real numbers) or one of its subsets is called a real function.

— Operations Of Functions –

1. Addition – Let f:X—->R and g:X—->R be any 2 functions and , then (f+g)(x) = f(x) + g(x),  .

• Subtraction – Let f:X—->R and g:X—->R be any 2 functions and , then (f-g)(x) = f(x) – g(x), .

• Multiplication — Let f:X—->R and g:X—->R be any 2 functions, then (fg)(x) = f(x)*g(x),.

• Division —  Let f:X—->R and g:X—->R be any 2 real functions, then (f/g)(x) = f(x)/g(x),

— Some functions –

1. Constant Function – It’s defined by f:R—->R and y=f(x)=c, each  and c is a constant.

2. Identity Function – It’s defined by f:R—->R, y = f(x) = x for each . Domain and range of f are R.

3. Polynomial Function – Defined by f:R—->R and y=f(x)=a₀+a₁x+a₂x₂₊ a₃x₃  . . . . . . . . . . . . a_nx_n

and  and a₀ , a_{1 ,} a₂ . . . . . .

• Rational Function – It’s in the form f(x)/g(x) where f(x) and g(x) are polynomial functions and g(x)
• The Modulus Function – It’s defined by f:R—->R where f(x) = [x].
• Signum Function – It’s defined by f:R—->R where — if x>0, then f(x)= 1, if x = 0, then f(x) = 0, if x>0 then f(x) = -1.

• Greatest integer function – It’s defined by f:R—->R where f(x) = [x] and