TNSSLC Mathematics Relations and Functions Study Notes

TNSSLC Mathematics Relations and Functions Study Notes

Gottfried Wilhelm Leibniz(also known as von Leibniz) was a prominent German mathematician, philosopher, physicist and inventor. He wrote extensively on 26 topics covering wide range of subjects among which were Geology, Medicine, Biology, Epidemiology, Paleontology, Psychology, Engineering, Philology, Sociology, Ethics, History, Politics, Law and Music Theory. In a manuscript Leibniz used the word “function” tomean any quantity varying from point to point of a curve. Leibniz provided the foundations of Formal Logic and Boolean Algebra, which are fundamental for modern day computers. For all his remarkable discoveries and contributions in various fields, Leibniz is hailed as “The Father of Applied Sciences”.

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 (Fig.1.1).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. Th us the fi rst 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”.

Definition

If A and B are two non-empty sets, then the set of all ordered pairs (a, b) such that a Î A, b Î B is called the Cartesian Product of A and B, and is denoted by A´B .

Thus, A´B = {(a,b) |a A,b B} .

(i) A × B is the set of all possible ordered pairs between the elements of A and B such that the fi rst coordinate is an element of A and the second coordinate is an element of B.

(ii) B × A is the set of all possible ordered pairs between the elements of A and B such that the fi rst coordinate is an element of B and the second coordinate is an element of A.

(iii) If a = b, then (a, b) = (b, a). The above two verified properties are called distributive property of cartesian product over union and intersection respectively. In fact, for any three sets A, B, C we have (i) A×(B C) = (A×B) (A×C) (ii) A×(B C) = (A×B) (A×C).

Definition

Let A and B be any two non-empty sets. A ‘relation’ R from A to B is a subset of A´B satisfying some specified conditions. If x Î A is related to y Î B through R , then we write it as x Ry. x Ry if and only if (x,y) Î  .

Here are various real life situations illustrating some special relations:

1. Consider the set A of all of your classmates; corresponding to each student, there is only one age.

2. You go to a shop to buy a book. If you take out a book, there is only one price corresponding to it; it does not have two prices corresponding to it. (of course, many books may have the same price).

3. You are aware of Boyle’s law. Corresponding to a given value of pressure P, there is only one value of volume V.

4. In Economics, the quantity demanded can be expressed as Q = 360−4P , where P is the price of the commodity. We see that for each value of P, there is only one value of Q. Thus the quantity demanded Q depend on the price P of the commodity.

Functions play very important role in the understanding of higher ideas in mathematics. They are basic tools to convert from one form to another form. In this sense, functions are widely applied in Engineering Sciences.