INFINITE SET: PROPERTIES, EXAMPLES - MATHS - 2024

An infinite set is understood to be that set in which the number of its elements is uncountable. That is, no matter how large the number of its elements may be, it is always possible to find more.

The most common example is the infinite set of natural numbers N. It doesn't matter how big the number is, since you can always get a bigger one in a process that has no end:

N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, ………………, 41, 42, 43, ………………………………………., 100, 101, ………………………, 126, 127, 128, ………………… ……………………}

Figure 1. Symbol of infinity. (pixabay)

The set of stars in the universe is surely immense, but it is not known for sure whether it is finite or infinite. In contrast to the number of planets in the solar system which is known to be a finite set.

Properties of the infinite set

Among the properties of infinite sets we can point out the following:

1- The union of two infinite sets gives rise to a new infinite set.

2- The union of a finite set with an infinite one gives rise to a new infinite set.

3- If the subset of a given set is infinite, then the original set is also infinite. The reciprocal statement is not true.

You cannot find a natural number capable of expressing the cardinality or number of elements of an infinite set. However, the German mathematician Georg Cantor introduced the concept of a transfinite number to refer to an infinite ordinal greater than any natural number.

Examples

The natural N

The most frequent example of an infinite set is that of natural numbers. The natural numbers are those that are used to count, however the whole numbers that may exist are uncountable.

The set of natural numbers does not include zero and is commonly denoted as the set N, which in extensive form is expressed as follows:

N = {1, 2, 3, 4, 5,….} And is clearly an infinite set.

An ellipsis is used to indicate that after one number, another follows and then another in an endless or endless process.

The set of natural numbers joined with the set that contains the number zero (0) is known as the set N +.

N + = {0, 1, 2, 3, 4, 5,….} Which is the result of the union of the infinite set N with the finite set O = {0}, resulting in the infinite set N +.

The integers Z

The set of integers Z is made up of natural numbers, natural numbers with a negative sign and zero.

The integers Z are considered an evolution with respect to the natural numbers N used originally and primitively in the counting process.

In the numerical set Z of integers zero is incorporated to count or count anything and negative numbers to account for extraction, loss or missing something.

To illustrate the idea, suppose a negative balance appears in the bank account. This means that the account is below zero and not only is the account empty but it has a missing or negative difference, which somehow has to be replaced by the bank.

In extensive form the infinite set Z of integers is written like this:

Z = {……., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, ……..}

The rationals Q

In the evolution of the process of counting, and exchanging things, goods or services, fractional or rational numbers appear.

For example, in the exchange of half a loaf with two apples, at the time of recording the transaction, it occurred to someone that half should be written as one divided or divided into two parts: ½. But half of half the bread would be recorded in the ledgers as follows: ½ / ½ = ¼.

It is clear that this process of division can be endless in theory, although in practice it is until the last particle of bread is reached.

The set of rational (or fractional) numbers is denoted as follows:

Q = {………, -3,…., -2,….., -1, ……, 0,….., 1, ……, 2,….., 3, ……..}

The ellipsis between the two whole numbers means that between those two numbers or values ​​there are infinite partitions or divisions. That is why the set of rational numbers is said to be infinitely dense. This is because no matter how close two rational numbers may be to each other, infinite values ​​can be found.

To illustrate the above, suppose that we are asked to find a rational number between 2 and 3. This number can be 2⅓, which is what is known as a mixed number consisting of 2 whole parts plus a third of the unit, which is equivalent to writing 4/3.

Between 2 and 2⅓ another value can be found, for example 2⅙. And between 2 and 2⅙ another value can be found, for example 2⅛. Between these two another, and between them another, another and another.

Figure 2. Infinite divisions in rational numbers. (wikimedia commons)

Irrational numbers I

There are numbers that cannot be written as the division or fraction of two whole numbers. It is this numerical set that is known as the set I of irrational numbers and it is also an infinite set.

Some notable elements or representatives of this numerical set are the number pi (π), the Euler number (e), the golden ratio or golden number (φ). These numbers can only be written roughly by a rational number:

π = 3.1415926535897932384626433832795 …… (and continues to infinity and beyond…)

e = 2.7182818284590452353602874713527 ……. (and continues beyond infinity…)

φ = 1.61803398874989484820 …….. (to infinity…..and beyond…..)

Other irrational numbers appear when trying to find solutions to very simple equations, for example the equation X ^ 2 = 2 does not have an exact rational solution. The exact solution is expressed by the following symbology: X = √2, which is read x equal to the root of two. An approximate rational (or decimal) expression for √2 is:

√2 ≈1.4142135623730950488016887242097.

There are countless irrational numbers, √3, √7, √11, 3 ^ (⅓), 5 ^ (⅖) to name a few.

The set of reals R

Real numbers are the number set most often used in mathematical calculus, physics, and engineering. This number set is the union of the rational numbers Q and the irrational numbers I:

R = Q U I

Infinity greater than infinity

Among the infinite sets some are greater than others. For example, the set of natural numbers N is infinite but is a subset of integers Z which is infinite, so infinite set Z is greater than the infinite set N.

Similarly, the set of integers Z is a subset of the real numbers R, and therefore the set R is "infinity" the infinite set Z.

References

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