Let A and B are two nonempty sets. The Cartesian product A × B of these sets is defined as the set f all ordered pairs (a, b) such that a ε A and b ε B. For example If A = {2, 3. 4} and B = {4, 9, 16} then A × B = {(2, 9), (2, 16), (3, 9), (3, 4), (3, 16), (4, 9), (4, 4), (4, 16)}. Out of these ordered paired elements of some are related with each other (In above example the second elements of a bold marked ordered pair is the square of the first. Such ordered pairs are called related ordered pairs.
A subset 'f' of the Cartesian product A X B is called a function from A to B if and only if to each 'a' ε A, there exists a unique 'b' in B such that (a, b) ε f. Thus the function from A to B can be described as the set of ordered pairs (a, b) such that a ε A and b ε B and for each 'a' there is a unique 'b'. This function may be written as:
f : A → B or A → B
Thus, a relation from A to B is a function if and only if
(i) To each a ε A, there exists a unique 'b' n B such that (a, b) ε f
(ii) (a_{1}, b_{1}) ε f and (a_{1}b_{2}), ε f => b_{1} = b_{2}.
Graphically, if f(x) is plotted in the y axis against x and if a line parallel to y axis cuts f(x) at more than one point then f(x) does not fulfill the requirement of a function, because for same value of x, you will have two values of y. (see fig. 1)
x = y^{2} is not a function, because of x = 4, y = +2.
e.g. If A = [a, b, c} and B = {d, e, f}
Then f = {(a, d), (b, d), (c, e)} is a function from A to B but g = {(a, d), (a, e)} is not a function from A to B.
The unique element b ε B assigned to a ε A is called the image of 'a' under f for value of f at 'a'. 'a' is called the preimage of 'b'. Also 'a' is called the independent variable, and 'b' is called the dependent variable.
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