Functions
A function is a relation between sets that associates to every element of a first set exactly one element of the second set. If A and B are two non-empty sets, then a function ƒ from A to B is a subset of A x B, with two important restrictions :
- for every a that belongs to A, (a, b) should belongs to ƒ for some b which belongs to B
i.e. ∀ a ∈ A, (a, b) ∈ ƒ for some b ∈ B. - if (a, b) belongs to ƒ and (a, c) also belongs to ƒ, then b and c should be equal
i.e. if (a, b) ∈ ƒ and (a, c) ∈ ƒ then b = c.
Note: There may be some elements of second set which are not related to any element of first set. But every element of first element must be related with only one element of second set. Also an element of second set can be related to any number of elements of first set.
The first set is called Domain of the function and the second set is called Co-Domain of the function.
A relation 'f' from a set 'A' to a set 'B' is called a function, if each element of A is mapped with a single element of set B.
If |A| = n and |B| = m, then total number of functions possible from A to B is mn and from B to A is nm.
Let ƒ denote a function from A to B.Then
ƒ : A → B
which can be readed as: "ƒ is a function from A into B", or "ƒ takes A into B", or "ƒ maps A into B".
Representation of a Function
The two sets A and B are represented by two ellipses. The function ƒ : A → B is represented by a collection of arrows joining the points which represent the elements of A and corresponds elements of B.
Domain of a function
Let ƒ be a function from A to B. The set A is called the domain of the function ƒ.
Co-Domain of a function
Let ƒ be a function from A to B. The set B is called co-domain of the function ƒ.
Range of a function
The range of a function is the set of picture of its domain. It is a subset of its co-domain which contains only those elements that are related by the elements of domain. It is denoted as ƒ(domain).
Example :
Let X = {1, 2, 3, 4}, Y = {a, b, c, d, e}
and ƒ = {(1, b), (2, a), (3, d), (4, c)}
Find the domain, co-domain and range of function.Domain of function : {1, 2, 3, 4}
Co-Domain of function : {a, b, c, d, e}
Range of function : {a, b, c, d}Let A = {a, b, c, d}, B = {r, s, t, u}
and ƒ = {(a, s), (b, u), (c, r), (d, s)}
Find the domain, co-domain and range of function. Also express the relation of each element as function.Domain of function : {a, b, c, d}
Co-Domain of function : {r, s, t, u}
Range of function : {r, s, u}
ƒ(a) = s, ƒ(b) = u, ƒ(c) = r and ƒ(d) = s
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