Basic Set Theory: Functions

Among the kinds of relations we have been talking about, there are some which are very useful in mathematics. A relation f \subset A \times B is called a function from A to B if

i) domain of f=A

ii) (a,b) \in f and (a,b^{'}) \in f \Longrightarrow b=b^{'}.

Thus a function from A to B is a relation whose domain is the whole of A and which is not one-to-many. The set B is called the codomain or range of the function f.

If f \subset A \times B is a function from A to B, then we write f: A \rightarrow B and say that f is a function from A to B. We also say that f maps A into B and often a function is called a map or mapping. If

(a,b) \in f, we write

b=f(a).

We call b the value of the function f at a or image of a under f. Observe that there can be no ambiguity in writing f(a) because it is impossible that (a,b) \in f and (a,b^{'}) \in f and b \neq b^{'}. So by definition of a function f: A \rightarrow B, we have for every a \in A a unique f(a) \in B. Thus, a function would be completely determined if we knew f(a) \in B for every a \in A. That is why sometimes a function is defined as a “rule” which associates to every element a \in A a unique element f(a) of B. We are obliged to put the word rule within quotation marks for the simple reason that the rule is often elusive. In fact, the aim in many situations is to “discover” the rule.

Examples

a) Suppose we are observing the position of a particle moving in a straight line. We know that at every instant of time the particle has a unique position on the line. If we agree that every point of the line corresponds to a real number with some convenient point on it decided to represent the real number zero, then the position of the particle at the time t is represented by the real number x(t). We observe that our with our ordinary notion of time and position, we do not expect a particle to occupy two different positions at the same time (nothing prevents, though,the particle from occupying the same position at two different times. Indeed, if the particle is at rest for a certain length of time, then it would have the same position for the entire length of time.) Suppose every time is represented by a real number, zero being a certain time deemed as the present time, then all future times are represented by positive real numbers and past times by negative real numbers. Here, a real number is used as an intuitive object, say, as points on a line, postponing a mathematical discussion of real numbers later. Let \Re be the set of real numbers, then the position of the particle moving in a straight line is, in fact, a function

x: \Re \rightarrow \Re

which associates to every time t, the position x(t).

b) Look at a railway time table. Below a particular train there are times recorded in a column. It records the times of arrival and departure of a train at certain stations. We may look upon this as a function whose domain consists of disjoint intervals of time and range consists of certain cities where the train stops. Thus, for every time in the mentioned interval we have a city where the train is at that time. The table does not say anything about the train’s position at a time after its departure and before its arrival at the next. This does not disqualify it to be a function if we take the domain to be the intervals of time depicting the times of arrival and departure at certain stations.

c) Consider the distance of a particle, falling freely under gravity, measured from the point from where the particle started its fall. if we record the time since it began to fall, we know that the distance x(t) at a time is given by the formula

x(t)=(1/2)gt^{2}

(assuming, of course that the gravity does not vary and there is no air resistance).

Here we have in fact a “rule” which tells us where the particle would be at any time. This certainly represents a function. However, we are often not so lucky to have a neat formula like this for other functions.

d) The incoming news on a particular day that tells us the temperature of the four metros recorded at 5.30 am on that day. Here, we have a definite temperature for every such metro at 5.30am of that day. So it is in fact a function whose domain consists of the set of four metros and range the real numbers quantifying temperature.

e) Look at your school time table. What does it record on a particular day? it records the subjects to be taught at different times of the day. A particular student has a particular subject being taught during a particular period,as a student is not expected to be in two different classes at the same time. So, for each student, the time table is a function whose domain is the set of periods and the range the set of subjects.

f) When we want to mail a letter enclosed in an envelope, we usually go to the post master with it to tell us the denomination of the stamp to be affixed. The post master weighs the letter and tells us the postage according to the weight. Here, we have a definite postage for a definite weight, we don’t have different rates for the same weight (for same kind of mail like registered or ordinary). The rate chart, with the post master, for a particular kind of mail, truly, is a function whose domain is the set of positive real numbers representing the weights of the mail and the range again consists of positive real numbers representing the postage. We write the chart as a function f such that

p-f(a)

meaning p is the postage to mail a letter of weight w.

g) We know that the solubility of a salt varies with temperature. That is to say that a given salt has a definite solubility at a definite temperature. So the relation which associates to every temperature the definite solubility of a salt is in fact a function.

h) Consider the population of a community of biological species at different times. If N(t) is the population of the community at time t, then surely

N: \Re \rightarrow \Re

represents a function.

i) If we measure the voltage of an AC power supply over a period of time, we observe that it is different at different times. At every instant of time, it has a voltage Sometimes, it is positive, sometimes it is negative and sometimes zero. In fact, for ordinary domestic supply in India, the voltage varies between 330 volts and -330 volts. But this change from 330 volts to -330 volts occurs within an interval of 1/100th of a second. (The ordinary DC voltmeters of your lab won’t be able to record this quick change in voltage). In fact, if V(t) is the volrage at time t, we have

V(t)=E_{0}\sin{(\omega t + \phi)}

where \omega=2\pi/50 and E_{0}=330 and \phi is a constant known as the phase.

j) Let S be a sphere resting on a horizontal plane H. Let n be the north pole and s be the southpole, where the sphere is touching the plane. Now let us join any other point p \in S to n and extend it to meet the plane at some point q. Now to every point p \in S - \{ n \}, we have a unique point q on H. This defines a function

f: S - \{ n \} \rightarrow H. This function is called the stereographic projection of the sphere on the plane.

k) The trivial map or function id: A \rightarrow A which sends every element of the set A to itself is called the identity map.

Look around, see and think…are there any functions that come to your notice? Let me know…

More later,

Nalin Pithwa

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