{"id":7917,"date":"2026-02-15T21:35:00","date_gmt":"2026-02-15T16:05:00","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=7917"},"modified":"2026-02-15T21:35:10","modified_gmt":"2026-02-15T16:05:10","slug":"relation-and-function-class-12-formula","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/relation-and-function-class-12-formula\/","title":{"rendered":"Relation and function class 12 formula"},"content":{"rendered":"\n

Here is the Relation and function class 12 formulas<\/p>\n\n\n\n

What is Relations<\/strong><\/p>\n\n\n\n

A relation R from a non-empty set A to a non-empty set B is a subset of the Cartesian product A \u00d7 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\u00d7 B)<\/p>\n\n\n\n

Domain of Relations<\/strong><\/p>\n\n\n\n

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<\/strong><\/p>\n\n\n\n

Range of Relations<\/strong><\/p>\n\n\n\n

The set of all the ending points is called the range of the relation R<\/strong><\/p>\n\n\n\n

Empty relation <\/strong><\/p>\n\n\n\n

it is the relation R in A given by $R = \\phi \\subset A \\times A$<\/p>\n\n\n\n

Universal relation<\/strong><\/p>\n\n\n\n

It is the relation R in A given by $R = A \\times A$.<\/p>\n\n\n\n

Reflexive relation<\/strong><\/p>\n\n\n\n

It is the Relation R in A with $(a, a) \\in R \\forall a \\in A$.<\/p>\n\n\n\n

i.e Relation R is reflexive if $(a,a) \\in R \\forall; a \\in A$<\/p>\n\n\n\n

Symmetric Relation<\/strong><\/p>\n\n\n\n

R in X is a relation satisfying $(a, b) \\in R$ implies $(b, a) \\in R$<\/p>\n\n\n\n

if $(a,b) \\in R$ then $(b,a) \\in R$ for all $a,b \\in A$<\/p>\n\n\n\n

Transitive relation<\/strong><\/p>\n\n\n\n

R in X is a relation satisfying $(a, b) \\in R$ and $(b, c) \\in R$ implies that $(a, c) \\in R$<\/p>\n\n\n\n

i.e $(a, b) \\in R$ , $(b, c) \\in R$ then $(a, c) \\in R$ for all $a,b,c \\in A$<\/p>\n\n\n\n

Equivalence relation<\/strong><\/p>\n\n\n\n

A relation which is reflexive ,symmetric and transitive all<\/p>\n\n\n\n

What is Functions<\/strong><\/p>\n\n\n\n

    \n
  1. A function is a “well-behaved” relation<\/li>\n\n\n\n
  2. 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.<\/li>\n<\/ol>\n\n\n\n

    One-one functions<\/strong><\/p>\n\n\n\n

    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<\/sub> and a2<\/sub> in A,
    if f(a1<\/sub>) = f(a2<\/sub>), then a1<\/sub> = a2<\/sub><\/p>\n\n\n\n

    Onto functions<\/strong><\/p>\n\n\n\n

    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<\/p>\n\n\n\n

    Bijective Functions<\/strong><\/p>\n\n\n\n

    A function which is both one-one and onto<\/p>\n\n\n\n

    Composition of functions<\/strong><\/p>\n\n\n\n

    The composition of functions f : A ->B and g : B ->C is the function
    $g \\circ f$ : A -> C given by $g \\circ f (x) = g(f(x))$ $\\forall x \\in A$<\/p>\n\n\n\n

    Composition of three Functions<\/strong><\/p>\n\n\n\n

    $h \\circ (g \\circ f) = (h \\circ g) \\circ f$<\/p>\n\n\n\n

    Invertible Functions<\/strong><\/p>\n\n\n\n

     A Function f : A-> B is invertible if we can find a function g: B- > A such that
    $f \\circ g=y$ and $g \\circ f=x$<\/p>\n\n\n\n

    Invertible functions are both one-one and onto functions<\/p>\n\n\n\n

    Invertibility of composition of the two functions<\/strong><\/p>\n\n\n\n

    if f and g are are invertible then <\/p>\n\n\n\n

    $g \\circ f$ is invertible and<\/p>\n\n\n\n

    $ (g \\circ f)^{-1} = f^{-1} \\circ g^{-1}$<\/p>\n\n\n\n

    Binary Operations<\/strong><\/p>\n\n\n\n

    A binary operation * on a set A is a function * : A x A -> A, we denote * (a,b) by a * b<\/p>\n\n\n\n

    Commutative Property on Binary Operations<\/strong><\/p>\n\n\n\n

    A binary operation * on the set X is called commutative, if a * b = b * a, for every $a, b \\in X$.<\/p>\n\n\n\n

    Associative Property on Binary Operations<\/strong><\/p>\n\n\n\n

    A binary operation * : A x A -> A is said to be associative if (a * b) * c = a * (b * c), for all $a, b, c, \\in A$<\/p>\n\n\n\n

    Identity Property<\/strong><\/p>\n\n\n\n

    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 $a \\in A$<\/p>\n\n\n\n

    Inverse of Binary Operations<\/strong><\/p>\n\n\n\n

    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<\/sup><\/p>\n\n\n\n

    Relations Articles<\/strong><\/p>\n\n\n\n