THE NATURE OF SETS

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Copyright © Cengage Learning. All rights reserved.

2.4 Finite and Infinite Sets

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Infinite Sets

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Infinite Sets

Certain sets, such as and A = {1000, 2000, 3000, . . .}, have a common property. Wecall these infinite sets. If the cardinality of a set is 0 or acounting number, we say the set is finite. Otherwise, wesay it is infinite. We can also say that a set is finite if it hasa cardinality less than some counting number, even thoughwe may not know its precise cardinality.

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Example 1 – Finding a one-to-one correspondence

Which of the following sets can be placed into a one-to-one correspondence?

K = {3, 4}, L = {4}, M = {3}, N = {three}, P = { }, Q = {t, h, r, e}

Solution:Notice that

L = {4} P = { }

M = {3} Q = { t , h , r , e }

N = {three}

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Example 1 – Solution

We see that L, M, and N all have the same cardinality, andusing that knowledge we see that any pair of them can beplaced into a one-to-one correspondence; we writeL M, L N, and M N.

We also see that P and Q can be placed into a one-to-one correspondence.

cont’d

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Infinite set

In the late 18th century, Georg Cantor assigned a cardinalNumber (pronounced “aleph-naught”) to the set of counting numbers. That is, is the cardinality of the set of counting numbers

= {1, 2, 3, 4, . . .}

The set

E = {2, 4, 6, 8, . . .}

also has cardinality , since it can be put into a one-to-one correspondence with set :

= {1, 2, 3, 4, . . ., n, . . .}

E = {2, 4, 6, 8, . . ., 2n, . . .}

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Infinite set

Notice that all elements of E are elements of the set , and that E . In such a case, we say that E is a proper subset of . Cantor used this property as a definingproperty for infinite sets.

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Example 2 – Infinite set

Show that the set of integers = {. . . , –3, –2, –1, 0, 1, 2, 3, . . .} is infinite.

Solution:We can show that the set of integers can be placed into a one-to-one correspondence with the set of counting numbers:

= {1, 2, 3, 4, 5, 6, 7, . . . , 2n, 2n + 1, . . .}

= {0, +1, –1, +2, –2, +3, –3, . . . ,+n, –n, . . .}

Since is a proper subset of , we see that the set of integers is infinite.

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Infinite set

If a set is finite or if it has cardinality , we say that the set is countable. Thus, both the set of counting numbers and the set of integers are countable.

A set that is not countable is said to be uncountable.In the example 2 we have seen that and are countable.

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Example 3 – Show that rationals are countable

Show that the set of rational numbers is countable.

Solution:

First, consider the positive rational numbers and arrange them in rows so that all the numbers with denominators of 1 are written in row 1, those with denominator 2 are written in row 2, and so on.

Notice that some numbers are highlighted because those numbers, when reduced, are found somewhere else in the list.

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Example 3 – Solution

We see that every nonnegative rational number will appear on this list. For example, 103/879 will appear in row 103 and column 879.

To set up a one-to-one correspondence between the set of nonnegative rationals and the set of counting numbers, we follow the path shown in next figure.

cont’d

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Example 3 – Solutioncont’d

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Example 3 – Solution

Now we set up the one-to-one correspondence:

(we skip because it is equal to and is already in our

list),

What we have shown is that the set of positive rational numbers has cardinality . Finally, we use the method shown in Example 2; that is, we let each negative number follow its positive counterpart, so we can extend this correspondence to include all negative rational numbers. Thus, the set of all rational numbers has cardinality and is therefore countable.

cont’d

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Example 4 – Show that reals are uncountable

Show that the set of real numbers is uncountable.

Solution:In order to prove that the set of real numbers isuncountable, we note that every set is either countable oruncountable.

We will assume that the set is countable, and then we willarrive at a contradiction. This contradiction, in turn, leadsus to the conclusion that our assumption is incorrect, thusestablishing that the set is uncountable. This process iscalled proof by contradiction.

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Example 4 – Solution

Let’s suppose that is countable. Then, there is some one-to-one correspondence between and , say:

Now, if we assume that there is a one-to-one correspondence between and , then every decimal number is in the above list.

cont’d

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Example 4 – Solution

To show this is not possible, we construct a new decimal as follows. The first digit of this new decimal is any digit different from the first digit of the entry corresponding to the first correspondence. (That is, anything other than 2 using the above-listed correspondence).

The second digit is any digit different from the second digit of the entry corresponding to the second correspondence (4 in this example).

Do the same for all the numbers in the one-to-onecorrespondence.

cont’d

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Example 4 – Solution

Because of the way we have constructed this new number, it is not on the list. But we began by assuming that all numbers are on the list (i.e., that all decimal numbers are part of the one-to-one correspondence).

Since both these statements cannot be true, the originalassumption has led to a contradiction. This forces us to accept the only possible alternative to the original assumption.

That is, it is not possible to set up a one-to-onecorrespondence between and , which means that is uncountable.

cont’d

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Infinite set

It can also be shown that the irrational numbers (real numbers that are not rational) are uncountable. Furthermore, the set {x | a x b}, where a and b are any real numbers, is also uncountable.

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Cartesian Product of Sets

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Cartesian Product of Sets

There is an operation of sets called the Cartesian productthat provides a way of generating new sets when given theelements of two sets. Suppose we have the sets

A = {a, b, c} and B = {1, 2}

The Cartesian product of sets A and B, denoted by A B,is the set of all ordered pairs (x, y) where x A and y B. For this example,

A B = {(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)}

Notice that A B B A.

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Example 5 – Cardinality of a Cartesian product

How many elements are in the Cartesian product of thegiven sets?

a. A = {Frank, George, Hazel} and

B = {Alfie, Bogie, Calvin, Doug, Ernie}

b. C = {U.S. Senators} and

D = {U.S. President, U.S. Vice President, Secretary of State}.

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Example 5(a) – Solution

One of the ways to find a Cartesian product is to represent the sets as an array.

Since a Cartesian product can be arranged as a rectangular array, we see that the number of elements is the product of the number of elements in the sets.

That is, since | A | = 3 and | B | = 5, we see |A B| = 3 5 = 15.

cont’d

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Example 5(b) – Solution

We are looking for |C D|, but we see it is not practical to form the rectangular array because | C | = 100 and | D | = 3. We still can visualize the size of the array even without writing it out, so we conclude

|C D| = | C | | D | = 100 3 = 300

cont’d

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Cartesian Product of Sets

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Example 6 – Classify as finite or infinite

Classify each of the sets as finite or infinite.

a. Set of people on Earth

b. Set of license plates that can be issued using three letters followed by three numerals

c. Set of drops of water in all the oceans of the world

Solution:

a. We do not know the size of this set, but we do know thatthere are fewer than 10 billion people on Earth. If the cardinality of a set is less than a finite number (and 10 billion is finite), then it must be a finite set.

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Example 6 – Solution

b. We can use the fundamental counting principle to calculate the number of possible license plates:

26 26 26 10 10 10 = 17,576,000

Note that each “26” counts the number of letters of the alphabet, and each “10” counts the number of numerals in each position. Since this is a particular number, we see that this set of license plates is finite.

c. There is a finite number of drops that make up an ounce, and there are a certain number of ounces in a gallon, and a certain number of gallons in a cubic foot.

cont’d

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Example 6 – Solution

The earth will fit entirely inside a cube of a certain size, and if this cube were filled with water, it would contain a finite number of gallons.

The number of drops of water in all the oceans of the world is certainly less than this number, and hence the set is finite.

cont’d