## FunctionsIt is a mapping in which every element of set A is uniquely associated at the element with set B. The set of A is called Domain of a function and set of B is called Co domain. ## Domain, Co-Domain, and Range of a Function:
Domain of function: {1, 2, 3, 4} Range of function: {a, b, c, d} Co-Domain of function: {a, b, c, d, e} ## Functions as a SetIf P and Q are two non-empty sets, then a function f from P to Q is a subset of P x Q, with two important restrictions - ∀ a ∈ P, (a, b) ∈ f for some b ∈ Q
- If (a, b) ∈ f and (a, c) ∈ f then b = c.
## Note1: There may be some elements of the Q which are not related to any element of set P.## 2. Every element of P must be related with at least one element of Q.
## Representation of a FunctionThe two sets P and Q are represented by two circles. The function f: P → Q is represented by a collection of arrows joining the points which represent the elements of P and corresponds elements of Q
Then f can be represented diagrammatically as follows
- f = {(x, 1), (y, 2), (z, 3), (k, 4)
- g = {(x, 1), (y, 1), (k, 4)
- h = {(x, 1), (x, 2), (x, 3), (x, 4)
- l = {(x, 1), (y, 1), (z, 1), (k, 1)}
- d = {(x, 1), (y, 2), (y, 3), (z, 4), (z, 4)}.
- It is a function. Range (f) = {1, 2, 3, 4}
- It is not a function because every element of X does not relate with some element of Y i.e., Z is not related with any element of Y.
- h is not a function because h (x) = {1, 2, 3, 4} i.e., element x has more than one image in set Y.
- d is not a function because d (y) = {2, 3} i.e., element y has more than image in set Y.
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