relations-functions

Types of relations

1. Empty relation - if am empty set is a subset of AxA then that empty set is called the empty relation on A.

2. Universal Relation - for any cartesian product, AxA, AxA is itself called universal relation on A

3. Identity Relation - If every element of A is related to itself only, then it the relation is called identity relation.

4. Reflexive Relation - relation R is said to be reflexive relation on A if $(a,a)\in R\forall a\in A$

5. Symmetric Relation - relation R is said to be reflexive relation on A if $(a,b)\in R\text{ and }(b,a)\in R\forall a,b\in A$

6. Transitive Relation - relation R is ALWAYS said to be transitive relation on A UNLESS $(a,b)\in R\text{ and }(b,c)\in R\text{ but }(a,c)\notin R\forall a,b,c\in A$

   IMPORTANT

   A relation on A is not transitive if and only if, it satisfies the above condition PERFECTLY

7. Equivalence Relation - relation R is said to be equivalence relation in A if it is reflexive, symmetric, and transitive in A

8. Antisymmetric Relation - a relation R is said to be antisymmetric if $(a,b)\in R\text{ and }(b,a)\in R\forall a,b\in A\text{only when a = b}$

   IMPORTANT

   Antisymmetric is NOT opposite to symmetric

   NOTE

   the greater than equal to relation is always Antisymmetric

Classification of Functions

1. Injective or one-one - If any two different inputs(domain elements), DON’T give the same output(range elements), then that function is injective function
2. many-one - If any two different inputs(domain elements), DO give the same output(range elements), then that function is many-one function
3. Surjective or onto - If codomain of a function is EQUAL to it’s range then that function is called surjective function
4. into - If codomain of a function is NOT EQUAL to it’s range then that function is called into function
5. bijective - A function is said to be bijective, if and only if, it is BOTH injective(one-one) and surjective(onto)

Inverse of a function

A function f(x) from A to B is said to be invertible, if there exists a function g(x) from B to A such that $fog(y)=f(g(y))=y\text{and}gof(x)=g(f(x))=x$ Then, g(x) is called inverse of f(x) and is denoted by $f^{-1}(x)$

IMPORTANT

A function is invertible i.e it’s inverse exists if and only if the function is bijective

In order to find the inverse of a function, just follow the following steps

1. Write y = f(x) and replace x with y
2. Find the value of y and that value will be $f^{-1}(x)$

IMPORTANT

If fog(x) = gof(x) then it means that g(x) is inverse of f(x)

NOTE

The solutions for $f^{-1}(x)=x$ are same as solutions for $f(x)=x$

Key Points

• Number of relations from set A to B containing n and m number of elements respectively are $2^{nm}$

• Number of relations in a set containing n number of elements, $2^{n^2}$

• Number of reflexive relations in a set containing n number of elements, $2^{n(n-1)}$

• Number of symmetric relations in a set containing n number of elements, $2^{n(n+1)/2}$

• Number of into functions from set A to B containing n and m number of elements respectively $n^m$

• Number of onto(surjective) functions from set A to B containing n and m number of elements respectively ${\sum }_{k=0}^n(-1{)}^k\left(\frac{n!}{k!(n-k)!}\right)(n-k{)}^m$

• Number of one-one(injective) functions from set A to B containing n and m number of elements respectively $\frac{n!}{(n-m)!}$

  IMPORTANT

  If m > n, then there are no injective functions

• Number of many-one functions from set A to B containing n and m number of elements respectively $n^m$

• If any function f(x) is from A to B and any function g(x) is defined from B to C, then

  • f(g(x)) or fog(x) will be defined from B to B
  • g(f(x)) or gof(x) will be defined from A to C

• For a non-zero function f(x) we have $f(xy)=f(x)+f(y)=>f(x)=K\ln (x)$ $f(xy)=f(x)\times f(y)=>f(x)=x^n$ $f(x+y)=f(x)+f(y)=>f(x)=Kx$ $f(x+y)=f(x)xf(y)=>f(x)=a^x$ $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})\text{ and f(x) is a polynomial}=>f(x)=\pm x^n+1$

Important Questions

1. If R is the relation “less than” from A = {1,2,3,4,5} to B = {1,4,5} then write the relation R in rooster form. Also find R^-1

2. If R = {(x,y): x,y belong to N, 2x + y = 41} is a relation on natural numbers i.e N then find the Domain and Range of the relation and also check whether it is Reflexive, Symmetric or Transitive

3. If R = {(x,y): 1 + xy > 0} is a relation on set of Real numbers, then check whether it is Reflexive, Symmetric or Transitive

4. Let X = {$4^n-3n-1:n\in \mathbb{N}$} $\mathbb{N}$ = Set of natural numbers

   Y = {$9(n-1):n\in \mathbb{N}$}

   Then, $X\cup Y$ = ? Options = (X,Y,N,Y-X)

5. Show that the following function f(x) defined from R to R is a neither injective nor surjective. $f(x)=\frac{x^2}{x^2+1}\forall x\in \mathbb{R}$

6. Let A = {1,2,3}. Find number of relations containing (1,2) and (1,3), which are reflexive and symmetric but not transitive.

7. Let A = {1,2,3}, then find the number of equivalence relations containing (1,2)

8. Find the range and inverse of following function $f(x)=\frac{4x}{3x+4}x\ne -\frac{4}{3}x\in \mathbb{R}$

9. Set A has 3 elements and set B has 4 elements. Then, find the number of injective mapping that can be defined from A to B.

10. Let f:R→R be a function defined by following, then find the pre-images of 17 and -3. $f(x)=x^2+1$

11. If set A contains 5 elements and set B contains 6 elements, then find the number of bijective mappings from A to B.

12. Let set X = {1,2,3} and a relation R is defined in X as R = {(1,3),(2,2),(3,2)} then find the minimum ordered pairs which should be added in relation R to make it reflexive and symmetric.

13. Relation R in set A = {1,2,3} is defined as R = {(1,1),(1,2),(2,2),(3,3)} then which ordered pair in R shall be removed to make it an equivalence relation in A ?

14. Let T be the set of triangles in the Euclidean plane and let a relation R on T be defined by following then check whether R is an equivalence relation or not $R=\left\{(a,b):\text{a is congruent to b}\forall a,b\in T\right\}$

15. The function P is defined as: to each person on the Earth is assigned a date of birth. Is this function injective ? Give reason.