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The Basic Concepts of Function and Probability
1.1 SETS AND FUNCTIONS
Definition 1. A set is any well-defined collection of objects. By "well-defined" we mean that we can tell what objects are in the collection and what objects are not in the collection. Any member of a set is called an element, or point, of the set.
Example 1. The collection of people who own a home within the city limits of Chicago is a set. Each person owning a home within the city limits of Chicago is an element of this set.
Example 2. If a group of students take an examination, then the collection of scores obtained by the students forms a set. Each individual score is an element of the set.
Example 3. A deck of data cards for use in a computer program forms a set. Each card is an element of the set.
Certain or all, of the elements in one set may be related in some way to certain, or all, of the elements of another set. This point is illustrated in the following examples.
Example 4. Let S be the set of all people. Then the phrase "is the parent of" relates each element of S, that is, each person, to those elements of S(persons) of which he is the parent. If x is a person who is not the parent of anyone, then "x is the parent of y" will not be satisfied for any person y.
Example 5. Let S be a set of students who took an examination and T be the set of scores obtained by the students. Then the phrase "has the score" assigns some element of T to each element of S.
Example 6. Let R be the set of real numbers. Then the rule f(x) = 2x3 assigns to each real number x another real number f(x) which is twice the cube of x.
Although the phrase "is the parent of" (Example 4) relates some people to no one at all, it also relates those who are the parents of several children to more than one person. In Example 6, however, not only does the rule f(x) = 2x3 relate each real number x to some real number, but x is related to aunique real number 2x3. Given x, there is no choice as to what f (x) is. When each element of one set is related to one and only one element of another set (which may also be the same as the first set), then we say that we have afunction from the first set into the second set. More formally, we make the following definition:
Definition 2.A rule,phrase,or relationship,which assigns to each element of a set S one and only one element of a set T is said to be a function from S into T.
Thus, the rule in Example 6 is a function, while the phrase in Example 4 does not give a function.
Consider Example 5 again. The phrase "has the score" is a function from S into T since each student has one and only one score. If a student s has a score t, then we may, if we wish, represent this fact by means of the "ordered pair" (s,t). More generally, we may represent the function "has the score" of Example 5 by all ordered pairs (s,t), where s is a student and s has the score t.
Definition 3. If S and T are any two sets, then any object of the form(s, t),where s is an element of S and t is an element of T, is said to be an ordered pair with s as its first coordinate and t as its second coordinate.
An ordered pair is a pair of elements, one from each of two sets, where the order in which the elements are given is important. An ordered pair is the opposite of an unordered pair, that is, a pair of elements, one from each of two sets, where the order is not taken into account. From the unordered pair containing, say 1 and 2, we can form two ordered pairs: (1, 2) and (2, 1).
If some rule relates each element of a set S to a unique element of a set T, then the rule gives rise to the set of all ordered pairs (s, t) such that s and t are elements of S and T, respectively, and t is related by the rule to s. Moreover, since no element of S is related to more than one element of T, but each element of S is related to some element of T, each element of S will appear as a first coordinate once and only once in the ordered pairs that the rule determines. The collection of ordered pairs determined by such a rule (function) is a subcollection of the set of all ordered pairs that can be formed with an element of S in the first coordinate and an element of T in the second coordinate. This inspires the following definition.
Definition 4. Let S and T be any sets. Then the Cartesian product, or simply the product, of S with T is defined to be the set of all ordered pairs(s, t)such that s and t are elements of S and T, respectively. We denote the product of S with T by S × T.
If f is a function from a set S into a set T, then f determines a particular kind of subcollection of S × T, specifically one in which each element of S appears as a first coordinate once and only once. On the other hand, if we begin with a subcollection of S × T having the property that each element ofS appears as a first coordinate exactly once, then this subcollection itself determines a function, namely, the function which relates the element s of S to that element t of T such that (s, t) is the only point of the subcollection which has s as a first coordinate. Functions, therefore, can be considered either as rules relating the elements of one set S to elements of a set T, or as a subcollection of S × T. In sum, we can say:
Characterization of functions in terms of ordered pairs.A function f from a set S into a set T is a collection of elements of S × T such that each element of S appears as a first coordinate in f once and only once. *
We may also indicate that (s, t) is an element of f by writing t = f(s).
Example 7. Suppose S and T both consist of the elements 1, 2, and 3. Braces {} are customarily used to set off the elements of a set. Hence an equivalent form of the first sentence of this example is "Suppose S = T = {1, 2, 3}."
Then S × T contains all the elements in the array shown in Table 1.
Each of the following are functions from S into T. Each element of S appears as a first coordinate once and only once in each function. Observe that this is equivalent to saying that each function contains one and only one element from each row of Table 1.
f1 = {(1, 1), (2,2), (3,3)},
f2 = {(1, 3), (2, 3), (3, 1)},
f3 = {(1, 1), (2, 1), (3, 1)}.
These are certainly not all the functions from S into T; in all, there are 27 such functions. Note that f1 can also be characterized by the rule f1(s) = s for each element s of S. The function f3 can be characterized by the rule f3(s) = 1 for each element s of S. We may also specify f2 by stating that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Example 8. Let R be the set of real numbers. Then R × R is the set of all ordered pairs (x,y), where x and y are both real numbers. The reader should recognize R × R as the ordinary coordinate plane. If the subset (= subcollection) f of R × R is a function, then each real number x must appear as a first coordinate in f once and only once. If x0 is a fixed, real number, then the line whose equation is x = x0 is a line parallel to the y -axis of the coordinate plane (Fig. 1). There is exactly one point (x0, f(x0)) of f whose first coordinate is x0; hence there is exactly one point of f on the line x = x0.
On the other hand, if f is a subset of R × R such that each line of the formx = x0 meets f in exactly one point, then f is a function. If some line x = x0does not meet f, then x0 does not appear as a first coordinate in f; and if some line x = x0 meets f in more than one point, then f contains at least two points with first coordinate x0.
Example 9. Let S and T be as in Example 7, and
g1 = {(1, 2), (1, 3), (2, 3), (3, 1)},
g2 = {(1, 3), (3, 2)}.
Then g1 and g2 are not functions from S into T. In the case of g1, the element 1 of S appears as a first coordinate twice; hence g1(1) is not clearly defined. On the other hand, 2 does not appear as a first coordinate in g2; hence g2(2) is not defined at all.
The following terminology is standard; we include it for the sake of completeness.
Definition 5. If f is a function from a set S into a set T, we call S the domain of f and T the range of f. The image of f is defined to be the set of all elements t of T such that t = f (s) for some element s of S.
Example 10. In Example 7, the domain and range are both {1, 2, 3} for each of the functions f1, f2, and f3. The image of f1 is {1, 2, 3}, of f2 is {1, 3}, and of f3 is {1}. The image of f is merely the set of elements of the range which appear as second coordinates in f.
EXERCISES
1. We have denoted certain sets by listing their elements between braces, for example f1 = {(1, 1), (2, 2), (3, 3)}. We might also have denoted a set by giving an arbitrary element of the set together with a condition that the element must satisfy to be in the set, all written in the following format: {x | condition that x must satisfy to be in the set}. Thus {y | y is a house} is the set of all houses. Express verbally each of the following sets.
a) {1, 45}
b) {a, b, 6, 7, 9}
c) {{1}}
d) {1, {1}}
e) {x | x is an animal}
f) {w | w is a citizen of Canada}
g) {y | y is an Indian and y lives in Iowa}
h) {z | z is an integer divisible by 2}
2. A set S is said to be a subset of a set T if each element of S is an element of T. Thus {1, 2} is a subset of {1, 2, 3}. A function f from a set S into a set T is a special kind of subset of S × T. We use S [subset] T to denote that S is a subset of T. Which of the following statements are true? If a statement is true, prove it; if it is false, try to find an instance in which the statement should apply, but does not. In the following, S, T, and W are sets.
a) If S [subset] T and T [subset] W, then S [subset] W.
b) If S [subset] W, then W [subset] S.
c) S [subset] S
d) If S [subset] T, but T is not a subset of W, then S is not a subset of W.
3. Write out all the elements of the following products. Try to arrange the elements in an array similar to that given in Table 1.
a) {1, 2} × {3, 4}
b) {a, b, c } × {6,7}
c) {q } × {t}
d) {1, 2, 3, 4} × {5, 6, 7, 8}
e) {A, B, C, D} × {A, B, C, D}
4. Find four distinct functions from S into T, where S and T are given as in each of the following. It may help to use the arrays constructed in Exercise 3. Compute the range, domain, and image of each function found.
a) S = {a, b, c} and T = {6, 7}
b) S = {1, 2, 3, 4} and T = {5, 6, 7, 8]
c) S = T = {A, B, C, D}
5. Let S be the set of living human beings. Which of the following phrases define a function from S into S? If a phrase fails to define a function, explain why it fails.
a) Is the cousin of
b) Is the father of
c) Is the same age as
d) Has as mother
If S were the set of all human beings who are living or have ever lived, would any of the phrases (a) through (d) define a function from S into S?
6. Which of the following define functions from the set R of real numbers into R? Indicate those functions whose image is the entire set of real numbers.
a) f(x) = x
b) g(y) = y + 2
c) h(w) = ±3w
d) f(x) = x1/2
e) g(u) = u2
f) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
1.2 THE NOTION OF PROBABILITY
The general purpose of probability theory is to make more "mathematical" such statements as "very likely" and "not much chance." Given some event E, we wish to assign a number to E, the probability of E, which will measure in some suitable fashion the chance that E will occur. We would also like to know how to manipulate probabilities once they have been assigned. For example, given the probabilities of the events E and E', we would like to be able to derive the probability of "either E occurs, or E' occurs."
We formulate the following definitions to help make the discussion more precise.
Definition 6. An experiment is a particular procedure to be performed, or a set of circumstances to be present simultaneously. The particular procedure to be performed, or the set of circumstances to be present, must be clearly defined.
A trial is one particular run of an experiment; that is, a trial is one actual performance of the procedure specified by an experiment, or a particular situation in which all the circumstances called for by the experiment are present.
A simple event is a possible outcome of a particular trial.
A sample space is the set of all simple events associated with an experiment.
The following examples illustrate the concepts presented in Definition 6.
Example 11. Taking a particular coin, tossing it into the air, and letting it come to rest on the floor is an experiment. A particular toss of the coin in accordance with the directions is a trial. The simple events associated with the experiment would be "heads" and "tails"; thus the sample space is {heads, tails}.
If the experiment were to flip the coin twice, then heads could occur on both the first and second tosses, or heads could occur on the first toss and tails on the second toss, etc. If H and T represent heads and tails, respectively, then the sample space for this experiment is
S = {HT, TH, HH, TT}.
In this latter experiment the occurrence of exactly one head is equivalent to the occurrence of one of the simple events HT and TH. We can associate the event "one occurrence of heads" with the subset {HT, TH } of the sample space S.
Definition 7. An event is any subset of a sample space.
Example 12. If the coin of Example 11 is flipped twice, then {HT}, {HH}, {HH, TT}, and S itself are all events. The event {HT, TH, HH} can be thought of as the "nonoccurrence of TT," or as "the occurrence of at least one head."
Two events A and B may be mutually exclusive. Informally, this means that A and B cannot occur together. In terms of a sample space, it means thatA and B share no simple events in common; that is, their intersection is empty.
Two events may also be independent, that is, the occurrence or nonoccurrence of one of the events in no way affects the occurrence or nonoccurrence of the other event. More will be said about mutually exclusive and independent events later.
We may also "build other events from two given events A and B." In particular, we make the following definition.
Excerpted from Calculus and Statistics by Michael C. Gemignani. Copyright © 1998 Michael C. Gemignani. Excerpted by permission of Dover Publications, Inc..
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Overview
Starting with the basic concepts of function and probability, the text addresses some specific probabilities and proceeds to surveys of random variables and graphs, the derivative, ...