Sets relations and function-maths

SETS : A Set is a collection of well-defined objects. When we say well-defined, we mean that there is a clear rule saying which objects are in the set. Sets are usually denoted by capital letters A,B,C etc. The members of a set are called its elements. The Elements of a set are usually denoted by small letters a,b,c, etc.

Finite and Infinite Sets

There was a mathematician called Georg Cantor who created a new branch of mathematics called set theory in the late 19th century. Set theory involves collections of numbers or objects. Here's a set:

{1,2,3,4,5}

This set consists of five elements, namely the first five natural numbers. Now consider the set:

{6,7,8,9,10}

Are these sets of the same size? Yes, they are. This is because they both have five elements. As we will see later, this method of comparing sizes does not work for all sets. An alternate method for comparing set sizes is to match elements of sets in a one-one fashion.

Think of a small child who wants to compare the number of marbles she has with her brother's collection. Let's say she doesn't know how to count beyond ten. She can still compare the sizes of their collections of marbles by lining up their marbles in two parallel lines. The line on the left contains her marbles while the one on the right contains her brother's. If each marble on the left is aligned with exactly one marble on the right, then they both have the same number of marbles.

We can use the same idea to compare infinite sets. If we can find a way to pair up one member of set A with one member of set B, and if there are no members of A without a partner in B and vice versa then we can say that set A and set B have the same number of members. Formally, two sets and are of the same size if there is a function such that for every in , we have in and moreover, for every in , there exists an in such that .

Example

Consider our previous example. We want to know if the sets

{1,2,3,4,5} and {6,7,8,9,10}

have the same size. We can create the following matching.

1 6

2 7

3 8

4 9

5 10

Example

Let Set N be all counting numbers. N is called the set of natural numbers. 1,2,3,4,5,6,... and so on. Let Set B be the negative numbers -1,-2,-3, ... and so on. Can the members of N and B be paired up? The formal way of saying this is "Can A and B be put into a one to one correspondence"?

Obviously the answer is yes. 1 in set N corresponds with -1 in B. Likewise:

N B

1 -1

2 -2

3 -3

and so on. Here, the one-one function that maps from A to B is .

So useful is the set of counting numbers that any set that can be put into a one to one correspondence with it is said to be countably infinite.

Example

The set of integers is the set containing all elements from the set N, the set B and the element 0. That is

{... -3,-2,-1, 0, 1, 2, 3, ...}

The set of integers is usually denoted by Z. Note that N the set of natural numbers is a subset of Z. All members of N are in Z, but not all members of Z are in N.

Is the set of integers countably infinite? In other words, can the set of integers be put in one-one correspondence with the set of all natural numbers?

Since the set N is contained in the set Z, we may be tempted to declare that these two sets are not of the same size. However, we can match them:

Z N

0 1

-1 2

1 3

-2 4

and so on. We can write this one-one correspondence as a function

f(x)={(x-1)/2 , when x is odd

{-x/2 , when x is even

We can verify that this function generates all the integers in Z from the natural numbers in N. Strange indeed! A subset of Z (namely the natural numbers) has the same size as Z itself! Infinite sets are not like ordinary finite sets. In fact this is sometimes used as a definition of an infinite set. An infinite set is any set which can be put into a one to one correspondence with at least one of its subsets . Rather than saying "The number of members" of a set, people sometimes use the word cardinality or cardinal value. Z and N are said to have the same cardinality.

Exercises

1. Is the number of even numbers the same as the natural numbers?
2. What about the number of square numbers?
3. Is the cardinality of positive even numbers less than 100 equal to the cardinality of natural numbers less than 100? Which set is bigger? How do you know?
4. In what ways do finite sets differ from infinite ones?