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Here is the Relation and function class 12 formulas
What is Relations
A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A × B. It “maps” elements of one set to another set. The subset is derived by describing a relationship between the first element and the second element of the ordered pair (A× B)
Domain of Relations
The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R
Range of Relations
The set of all the ending points is called the range of the relation R
Empty relation
it is the relation R in A given by $R = \phi \subset A \times A$
Universal relation
It is the relation R in A given by $R = A \times A$.
Reflexive relation
It is the Relation R in A with $(a, a) \in R \forall a \in A$.
i.e Relation R is reflexive if $(a,a) \in R \forall; a \in A$
Symmetric Relation
R in X is a relation satisfying $(a, b) \in R$ implies $(b, a) \in R$
if $(a,b) \in R$ then $(b,a) \in R$ for all $a,b \in A$
Transitive relation
R in X is a relation satisfying $(a, b) \in R$ and $(b, c) \in R$ implies that $(a, c) \in R$
i.e $(a, b) \in R$ , $(b, c) \in R$ then $(a, c) \in R$ R for all $a,b,c \in A$
Equivalence relation
A relation which is reflexive ,symmetric and transitive all
What is Functions
- A function is a “well-behaved” relation
- A function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element.
One-one functions
A function from a set A to a set B, f is called a one-to-one function or injection, if, and only if, for all elements a1 and a2 in A,
if f(a1) = f(a2), then a1 = a2
Onto functions
A function f: A-> B is said to be onto(surjective) if every element of B is the image of some element of A under f, i,e
for every $y \in B$, there exists a element x in A where f(x)=y
Bijective Functions
A function which is both one-one and onto
Composition of functions
The composition of functions f : A ->B and g : B ->C is the function
$g \circ f$ : A -> C given by $g \circ f (x) = g(f(x))$ $\forall x \in A$
Composition of three Functions
$h \circ (g \circ f) = (h \circ g) \circ f$
Invertible Functions
A Function f : A-> B is invertible if we can find a function g: B- > A such that
$f \circ g=y$ and $g \circ f=x$
Invertible functions are both one-one and onto functions
Invertibility of composition of the two functions
if f and g are are invertible then
$g \circ f$ is invertible and
$ (g \circ f)^{-1} = f^{-1} \circ g^{-1}$
Binary Operations
A binary operation * on a set A is a function * : A x A -> A, we denote * (a,b) by a * b
Commutative Property on Binary Operations
A binary operation * on the set X is called commutative, if a * b = b * a, for every $a, b \in X$.
Associative Property on Binary Operations
A binary operation * : A x A -> A is said to be associative if (a * b) * c = a * (b * c), for all $a, b, c, \in A$
Identity Property
Given a binary operation * : A x A -> A, an element $e \in A$, if it exists, is called identity for the operation *, if a * e = a = e * a, for all $a \in A$
Inverse of Binary Operations
Given a binary operation * : A x A -> A with the identity element e in A, an element $a \in A$ is said to be invertible with respect to the operation *, if there exists an element b in A such that a * b = e = b * a and b is called the inverse of a and is denoted by a-1
Relations Articles
