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The A to Z of mathematics a basic guide - Sidebotham T.H.

Sidebotham T.H. The A to Z of mathematics a basic guide - Wiley publishing , 2002. - 489 p.
ISBN 0-471-15045-2
Download (direct link): theatozofmath2002.pdf
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♦ Function
♦ Mapping
118 CORRESPONDENCE
A relation, which is also known as a relationship, is simply defined to be a set of ordered pairs (x, y). A relation defines a connection between the first member x of each ordered pair and its corresponding second member y. The set of first numbers x is called the domain of the relation, and the set of second numbers y is called the range of the relation. The relation between x and y can be expressed in four ways:
♦ As a set of ordered pairs
♦ As an arrow graph
♦ As an equation
♦ As a Cartesian graph
1. A relation as a set of ordered pairs. If Luke has $2 more than Liz, and Liz has $2 more than Harry, then a relation between the members of the set {Luke, Liz, and Harry} could be “has $2 more than.” The set of ordered pairs for this relation is
R = {(Luke, Liz) (Liz, Harry)}
where R stands for the relation “has $2 more than.”
Another example of a relation expressed as a set of ordered pairs is the following: Suppose the relation “is two more than” exists between the set of counting numbers less than 10. This relation can be expressed as the set of ordered pairs
R = {(3,1) (4, 2) (5,3) (6,4) (7, 5) (8, 6) (9,7)}
The domain of R = {3,4, 5, 6,7, 8,9} and the range of R = {1, 2,3,4, 5, 6,7}.
2. A relation as an arrow diagram. The relation “is two more than” can be expressed as an arrow diagram, using the ordered pairs (see figure a).
3. A relation as an equation. The equation for the relation “is two more than” is
y = x — 2
where x is the set of counting numbers {3,4, 5, 6,7, 8, 9}. When writing the relation as an equation it is usual to list the domain at the same time so that we know the numbers for which the equation is defined.
CORRESPONDENCE
119
"t-
2 4 6 8 x
(b)
4. A relation as a Cartesian graph. The ordered pairs of the relation can be plotted as points on Cartesian axes, and it can be seen in figure b that they lie on a straight-line graph. The points cannot be joined up, because the numbers for x were stated as counting numbers. If the numbers for x and y were real numbers, the points could be joined up to make a continuous straight line.
Having explained the four different ways of expressing a relation, we now discuss the meaning of the word correspondence. There are four basic types of relations and each one is called a correspondence. These four types of relations are:
♦ One-to-one correspondence
♦ Many-to-one correspondence
♦ One-to-many correspondence
♦ Many-to-many correspondence.
6
4
2
Examples of each of these four correspondences are given below as arrow diagrams and ordered pairs.
1. One-to-one. “Is the wife of’ is an example of a one-one relation (see figure c). Helen is the wife of Norman, Margy is the wife of Harry, and Pat is the wife of Tom. The ordered pairs are {(Helen, Norman) (Margy, Harry) (Pat, Tom)}. In this correspondence, one member of the domain maps onto only one member of the range, and each member of the domain and range occurs only once. Therefore, it is called a one-to-one correspondence.
120 CORRESPONDENCE
2. Many-to-one. “Is the son of ” is an example of a many-one relation (see figure d). Paul is the son of Anne, Mark is the son of Anne, and John is the son of Ken. The ordered pairs are {(Paul, Anne) (Mark, Anne) (John, Ken)}. In this correspondence none of the members of the domain are repeated, but at least one member of the range is repeated, which is Anne. Therefore, it is called a many-to-one correspondence.
3. One-to-many. “Is the parent of” is an example of a one-many relation (see figure e). Anne is the parent of Paul, Anne is the parent of Mark, and Ken is the parent of John. The ordered pairs are {(Anne, Paul) (Anne, Mark) ( Ken, John)}. In this correspondence at least one member of the domain is repeated, which is Anne, but none of the members of the range is repeated. Therefore, it is called a one-to-many correspondence.
4. Many-to-many. “Is the brother of” is an example of a many-many relation (see figure f). John is the brother of Harry, John is the brother of Helen, Bill is the brother of Harry, and Bill is the brother of Helen. The ordered pairs are {(John, Harry) (John, Helen) (Bill, Harry) (Bill, Helen)}. In this correspondence at least one of the members of the domain is repeated, and at least one of the members of the range is repeated. Both John and Bill of the domain are repeated, and both Harry and Helen in the range are repeated. Therefore, it is called a many-to-many correspondence.
(f)
CORRESPONDING SIDES 121
The next term to explain is a junction. When a correspondence is one-to-one or a many-to-one, the relation is called a function. A function is sometimes called a mapping. If a relation is written as a set of ordered pairs, it is easy to recognize a function, because no members of the domain are repeated.
Example 1. Is the relation R = {(1,2) (2, 3) (3,2) (5,1)} a function?
Solution. Yes, because each member of the domain {1, 2, 3, 5} is not repeated in the domain of the ordered pairs.
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