Relations and Functions Introduction: The notion of sets provides the stimulus for learning higher concepts in mathematics. A set is a collection of well-defined distinguishable objects. This means that a set is merely a collection of something which we may recognize. In this chapter, we try to extend the concept of sets in two forms called Relations and Functions. For doing this, we need to first know about cartesian products that can be defined between two non-empty sets.
It is quite interesting to note that most of the day-to-day situations can be represented mathematically either through a relation or a function. For example, the distance travelled by a vehicle in given time can be represented as a function. The price of a commodity can be expressed as a function in terms of its demand. The area of polygons and volume of common objects like circle, right circular cone, right circular cylinder, sphere can be expressed as a function with one or more variables.
In class IX, we had studied the concept of sets. We have also seen how to form new sets from the given sets by taking union, intersection and complementation. Now we are about to study a new set called “cartesian product” for the given sets A and B.
Ordered Pair: Observe the seating plan in an auditorium. To help orderly occupation of seats, tokens with numbers such as (1,5), (7,16), (3,4), (10,12) etc. are issued. Th e person who gets (4,10) will go to row 4 and occupy the 10th seat.
Thus the first number denotes the row and the second number, the seat. Which seat will the visitor with token (5,9) occupy? Can he go to 9th row and take the 5th seat? Do (9,5) and (5,9) refer to the same location? No, certainly! What can you say about the tokens (2,3), (6,3) and (10,3)?
This is one example where a pair of numbers, written in a particular order, precisely indicates a location. Such a number pair is called an ordered pair of numbers. This notion is skillfully used to mathematize the concept of a “Relation”.
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