15 julio, 2024

Infinite set: properties, examples

It is understood by infinite set 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 of an infinite set is that of the natural numbers. No.. It doesn’t matter how big the number is, since you can always get a bigger one in a process that has no end:

No. = { 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,…………………… …………………}

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

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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.

It is not possible to 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 naturals N

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

The set of natural numbers does not include zero and is commonly denoted as the set No.which in extensive form is expressed as follows:

No. = { 1, 2, 3, 4, 5, ….} and is clearly an infinite set.

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

The set of natural numbers united 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 No. with the finite set EITHER = { 0 }, resulting in the infinite set N+.

The integers Z

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

whole numbers z are considered an evolution with respect to the natural numbers No. used originally and primitively in the process of counting.

In the number set z of the integers, zero is incorporated to count or count nothing and negative numbers to count extraction, loss or missing something.

To illustrate the idea, suppose that the bank account shows a negative balance. This means that the account is below zero and not only is the account empty, but it also has a missing or negative difference, which somehow has to be replenished to 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, when recording the transaction, someone thought that half should be written as one divided or sectioned into two parts: ½. But half of the half of 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 integers means that between those two numbers or values ​​there are infinite partitions or divisions. Therefore, 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, infinitely many values ​​can be found.

To illustrate the above, suppose 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 integer parts plus one 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.

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 (and), the golden ratio or golden number (φ). These numbers can only be written approximately by a rational number:

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

and = 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 means of the following symbols: X = √2, which is read as x equals to the root of two. An approximate rational (or decimal) expression of √2 is:

√2 ≈1.4142135623730950488016887242097.

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

The set of real numbers R

The real numbers are the number set most frequently used in mathematical calculation, physics, and engineering. This numerical set is the union of rational numbers Q and the irrational numbers Yo:

R. = Q OR Yo

infinity greater than infinity

Among infinite sets some are greater than others. For example, the set of natural numbers No. is infinite, however it is a subset of the integers z which is also infinite, therefore the infinite set z is greater than the infinite set No..

Similarly, the set of integers z is a subset of the real numbers R.and therefore the set R. is “more infinite” than the infinite set z.

References

famous. Examples of infinite sets. Retrieved from: celeberrima.com
Sources, A. (2016). BASIC MATH. An Introduction to Calculus. lulu.com.
Garo, M. (2014). Mathematics: quadratic equations: How to solve a quadratic equation. Marilu Garo.
Haeussler, EF, & Paul, RS (2003). Mathematics for administration and economics. Pearson Education.
Jiménez, J., Rodríguez, M., Estrada, R. (2005). Math 1 SEP. Threshold.
Precious, C.T. (2005). Mathematics course 3rd. Editorial Progress.
Rock, NM (2006). Algebra I Is Easy! So Easy. Team Rock Press.
Sullivan, J. (2006). Algebra and Trigonometry. Pearson Education.
Wikipedia. Infinite set. Recovered from: en.wikipedia.com

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