5
An element a X is said to be invertible with respect to the operation *, if there exists an element b X such that a*b = b*a = e, b X
If a*(b c) = (a*b) (a*c) a, b, c A (distribution of * from left) and (b c)*a = (b*a) (c*a) a, b, c A (distribution of * from right)
e X, if a*e = e*a = a, a X
If a*(b*c) = (a*b)* c, a,b, c X
For X is any set, if a*b = b*a, a, b, X
A binary operation '*' on a set A is a function * : AĂ—A A denoted by a* b i.e. a,b, A.
d
A function is said to be invertible iff it is both one-one and onto i.e. bijective Domain f B A {a, b, c, d} 1 a odomain 2 b {1, 2, 3, 4, 5,} C 3 c 4 Range {1, 3, 5}
A function f : x y is invertible, if a function g : y x such that gof = Ix and fog = Iy. Then, g is the inverse of f and it is denoted by f –1
It is a function that assigns a each member of its domain
f
B 1 2 3
If distinct elements of A have distinct images inB.
A a b c
Set B is the codomain of R e.g. B ={4, 5, 6, 7}
Relations and Functions
Set of second components of ordered The composition of functions f : A B and pairs e.g. {4, 7, 5} is the range. g : B C is denoted by gof, and is defined as gof : A C given by gof(x) = g(f(x)) x Set of first components of ordered pairs e.g. Let A = {4, 5, 6} and B = {4, 5, 6, 7} then {4, 5, 6} is the domain. R = {(4, 4), (5, 7), (6, 5)} is a relation in AĂ—B
A a b c d e
Two or more elements in A have the same image in B
If the set B is not entirely used up i.e. there exists at least one element in B that does not have pre image in A.
f
A a b c
A a b c d e
1 2 3 4
B
f
f
B 5 7 9
1 2 3 4
B
R is an equivalence relation, just in case R is reflexive, symmetric and transitive. Let A = {1, 2, 3}, R = {(1, 2), (1, 1), (2, 1), (2, 2), (3, 3)}. Here R is reflexive, symmetric and transitive, so R is an equivalence relation.
A relation R : A A is transitive if aRb, bRc aRc a,b,c A
A relation R : A A is symmetric if aRb bRa a, b A
A relation R : A A is reflexive if aRa a A
A relation R : A A is universal if a R b a, b A, R = AĂ—A.
A relation R : A A is empty if a R b a, b A, R = AĂ—A. For e.g. R ={(a,b) : a = b2}, A = {1, 5, 10}
If every element of A is related to itself only i.e. IA = {(a, a) : a = A}
If every element of B has pre-image in A, i.e. the set B is entirely used up i.e. if f(A) = B
A Relation R in a set A is a subset of AĂ—A. i.e, R AĂ—A.
Chapter - 1 Relations and Functions