Relations

RELATIONS

1.Indtroduction :
The role of relations in our daily life is very important where each relation has its own significance. For example:
  i) Relation of mother and son.
  ii) Relation of wife and Husband
Similarly, in mathematics also, there is a variety of relations, whose knowledge is crucial. Here, also each relation has its own meaning and significance. Let us understand this with the help of following examples.
i) 4 is square of 2 \( \Rightarrow \) Relation between 2 and 4
ii) \( \sin \theta = \frac{1} {{\cos ec\theta }} \Rightarrow \) Relation between \( \sin \theta \) and \( \cos ec\theta \) 
In all of the above examples, we conclude that every relation involves a pair of objects in a particular order. In this chapter, we’ll study about these ordered pairs and mainly about relations and functions in mathematics.

2. Ordered Pairs :
If a pair of elements is listed in a specific order, then such a pair is called an ordered pair. This ordered pair is written by listing the two objects in the specified order, separated by comma and enclosing the pair in parenthesis
Eg: The ordered pair of two elements a and b is donated by (a, b): a being first element and b is second element.
Note 1: 1) Two ordered pairs are equal if their corresponding elements are equal.
Eg: (a, b) = (c, d)\( \Rightarrow \)a = c and b = d
2) Remember (a,b)\( \ne \)(b,a)

As we know that graphically the ordered pair (2, 3) means that abscissa, x = 2 and ordinate, y = 3 
Thus from the graph it is obvious that ordered pairs (2, 3) and (3, 2) represent two different points and hence they are not equal
Ex:\( f\left( {\frac{x} {3} + 1,y - \frac{2} {3}} \right) = \left( {\frac{5} {3},\frac{1} {3}} \right) \) find the values of x and y
Sol: We have , \( \left( {\frac{x} {3} + 1,y - \frac{2} {3}} \right) = \left( {\frac{5} {3},\frac{1} {3}} \right) \Rightarrow \frac{x} {3} + 1 = \frac{5} {3} \Rightarrow \frac{x} {3} = \frac{5} {3} - 1 \Rightarrow x = \frac{{5 - 3}} {3} = \frac{2} {3} \) 
\( \Rightarrow y - \frac{2} {3} = \frac{1} {3} \Rightarrow y = \frac{1} {3} + \frac{2} {3} \Rightarrow y = 1\therefore x = 2,y = 1 \)