Biological immortality

From Wikipedia, the free encyclopedia

 

Biological immortality (sometimes referred to as bio-indefinite mortality) is a state in which the rate of mortality from senescence is stable or decreasing, thus decoupling it from chronological age. Various unicellular and multicellular species, including some vertebrates, achieve this state either throughout their existence or after living long enough. A biologically immortal living being can still die from means other than senescence, such as through injury, disease, or lack of available resources.

This definition of immortality has been challenged in the Handbook of the Biology of Aging, because the increase in rate of mortality as a function of chronological age may be negligible at extremely old ages, an idea referred to as the late-life mortality plateau. The rate of mortality may cease to increase in old age, but in most cases that rate is typically very high.

The term is also used by biologists to describe cells that are not subject to the Hayflick limit on how many times they can divide.

Cell lines

Biologists chose the word "immortal" to designate cells that are not subject to the Hayflick limit, the point at which cells can no longer divide due to DNA damage or shortened telomeres. Prior to Leonard Hayflick's theory, Alexis Carrel hypothesized that all normal somatic cells were immortal.

The term "immortalization" was first applied to cancer cells that expressed the telomere-lengthening enzyme telomerase, and thereby avoided apoptosis—i.e. cell death caused by intracellular mechanisms. Among the most commonly used cell lines are HeLa and Jurkat, both of which are immortalized cancer cell lines. HeLa cells originated from a sample of cervical cancer taken from Henrietta Lacks in 1951. These cells have been and still are widely used in biological research such as creation of the polio vaccine, sex hormone steroid research, and cell metabolism. Normal stem cells and germ cells can also be said to be immortal (when humans refer to the cell line).

Immortal cell lines of cancer cells can be created by induction of oncogenes or loss of tumor suppressor genes. One way to induce immortality is through viral-mediated induction of the large T‑antigen, commonly introduced through simian virus 40 (SV-40).

Organisms

According to the Animal Aging and Longevity Database, the list of organisms with negligible aging (along with estimated longevity in the wild) includes:

• Blanding's turtle (Emydoidea blandingii) – 77 years
• Olm (Proteus anguinus) – 102 years
• Eastern box turtle (Terrapene carolina) – 138 years
• Red sea urchin (Strongylocentrotus franciscanus) – 200 years
• Rougheye rockfish (Sebastes aleutianus) – 205 years
• Ocean quahog clam (Arctica islandica) – 507 years

In 2018, scientists working for Calico, a company owned by Alphabet, published a paper in the journal eLife which presents possible evidence that Heterocephalus glaber (Naked mole rat) do not face increased mortality risk due to aging.

Bacteria and some yeast

Many unicellular organisms age: as time passes, they divide more slowly and ultimately die. Asymmetrically dividing bacteria and yeast also age. However, symmetrically dividing bacteria and yeast can be biologically immortal under ideal growing conditions. In these conditions, when a cell splits symmetrically to produce two daughter cells, the process of cell division can restore the cell to a youthful state. However, if the parent asymmetrically buds off a daughter only the daughter is reset to the youthful state—the parent isn't restored and will go on to age and die. In a similar manner stem cells and gametes can be regarded as "immortal".

Hydra

Hydra

Hydras are a genus of the Cnidaria phylum. All cnidarians can regenerate, allowing them to recover from injury and to reproduce asexually. Hydras are simple, freshwater animals possessing radial symmetry and no post-mitotic cells. All hydra cells continually divide.  It has been suggested that hydras do not undergo senescence, and, as such, are biologically immortal. In a four-year study, 3 cohorts of hydra did not show an increase in mortality with age. It is possible that these animals live much longer, considering that they reach maturity in 5 to 10 days. However, this does not explain how hydras are consequently able to maintain telomere lengths.

Jellyfish

Turritopsis dohrnii, or Turritopsis nutricula, is a small (5 millimeters (0.20 in)) species of jellyfish that uses transdifferentiation to replenish cells after sexual reproduction. This cycle can repeat indefinitely, potentially rendering it biologically immortal. This organism originated in the Caribbean sea, but has now spread around the world.

  Similar cases include hydrozoan Laodicea undulata and scyphozoan Aurelia sp.1.

Lobsters

Research suggests that lobsters may not slow down, weaken, or lose fertility with age, and that older lobsters may be more fertile than younger lobsters. This does not however make them immortal in the traditional sense, as they are significantly more likely to die at a shell moult the older they get (as detailed below).

Their longevity may be due to telomerase, an enzyme that repairs long repetitive sections of DNA sequences at the ends of chromosomes, referred to as telomeres. Telomerase is expressed by most vertebrates during embryonic stages but is generally absent from adult stages of life. However, unlike vertebrates, lobsters express telomerase as adults through most tissue, which has been suggested to be related to their longevity. Contrary to popular belief, lobsters are not immortal. Lobsters grow by moulting which requires a lot of energy, and the larger the shell the more energy is required. Eventually, the lobster will die from exhaustion during a moult. Older lobsters are also known to stop moulting, which means that the shell will eventually become damaged, infected, or fall apart and they die. The European lobster has an average life span of 31 years for males and 54 years for females.

Planarian flatworms

Polycelis felina, a freshwater planarian

Planarian flatworms have both sexually and asexually reproducing types. Studies on genus Schmidtea mediterranea suggest these planarians appear to regenerate (i.e. heal) indefinitely, and asexual individuals have an "apparently limitless [telomere] regenerative capacity fueled by a population of highly proliferative adult stem cells". "Both asexual and sexual animals display age-related decline in telomere length; however, asexual animals are able to maintain telomere lengths somatically (i.e. during reproduction by fission or when regeneration is induced by amputation), whereas sexual animals restore telomeres by extension during sexual reproduction or during embryogenesis like other sexual species. Homeostatic telomerase activity observed in both asexual and sexual animals is not sufficient to maintain telomere length, whereas the increased activity in regenerating asexuals is sufficient to renew telomere length... "

Lifespan: For sexually reproducing planaria: "the lifespan of individual planarian can be as long as 3 years, likely due to the ability of neoblasts to constantly replace aging cells". Whereas for asexually reproducing planaria: "individual animals in clonal lines of some planarian species replicating by fission have been maintained for over 15 years". They are "literally immortal."

Attempts to engineer biological immortality in humans

Although the premise that biological aging can be halted or reversed by foreseeable technology remains controversial, research into developing possible therapeutic interventions is underway. Among the principal drivers of international collaboration in such research is the SENS Research Foundation, a non-profit organization that advocates a number of what it claims are plausible research pathways that might lead to engineered negligible senescence in humans.

In 2015, Elizabeth Parrish, CEO of BioViva, treated herself using gene therapy with the goal of not just halting, but reversing aging.

For several decades, researchers have also pursued various forms of suspended animation as a means by which to indefinitely extend mammalian lifespan. Some scientists have voiced support for the feasibility of the cryopreservation of humans, known as cryonics. Cryonics is predicated on the concept that some people considered clinically dead by today's medicolegal standards are not actually dead according to information-theoretic death and can, in principle, be resuscitated given sufficient technological advances. The goal of current cryonics procedures is tissue vitrification, a technique first used to reversibly cryopreserve a viable whole organ in 2005.

Similar proposals involving suspended animation include chemical brain preservation. The non-profit Brain Preservation Foundation offers a cash prize valued at over $100,000 for demonstrations of techniques that would allow for high-fidelity, long-term storage of a mammalian brain.

In 2016, scientists at the Buck Institute for Research on Aging and the Mayo Clinic employed genetic and pharmacological approaches to ablate pro-aging senescent cells, extending healthy lifespan of mice by over 25%. The startup Unity Biotechnology is further developing this strategy in human clinical trials.

In early 2017, Harvard scientists headed by biologist David Sinclair announced they have tested a metabolic precursor that increases NAD+ levels in mice and have successfully reversed the cellular aging process and can protect the DNA from future damage. "The old mouse and young mouse cells are indistinguishable", David was quoted. Human trials were planned to begin shortly in what the team expect is 6 months at Brigham and Women's Hospital, in Boston.

In the September 2019 article, a group of scientists reported successfully reversing the epigenetic aging in humans.

Criticism

To achieve the more limited goal of halting the increase in mortality rate with age, a solution must be found to the fact that any intervention to remove senescent cells that creates competition among cells will increase age-related mortality from cancer.

Immortalism and immortality as a movement

In 2012 in Russia, and then in the United States, Israel, and the Netherlands, pro-immortality transhumanist political parties were launched. They aim to provide political support to anti-aging and radical life extension research and technologies and want to ensure the fastest possible—and at the same time, the least disruptive—societal transition to radical life extension, life without aging, and ultimately, immortality. They aim to make it possible to provide access to such technologies to the majority of people alive today.

Future medicine, life extension and "swallowing the doctor"

Future advances in nanomedicine could give rise to life extension through the repair of many processes thought to be responsible for aging. K. Eric Drexler, one of the founders of nanotechnology, postulated cell repair devices, including ones operating within cells and using as yet hypothetical molecular machines, in his 1986 book Engines of Creation. Raymond Kurzweil, a futurist and transhumanist, stated in his 2005 book The Singularity Is Near that he believes that advanced medical nanorobotics could completely remedy the effects of aging by 2030. According to Richard Feynman, it was his former graduate student and collaborator Albert Hibbs who originally suggested to him in around 1959 the idea of a medical use for Feynman's theoretical micromachines. Hibbs suggested that certain repair machines might one day be reduced in size to the point that it would, in theory, be possible to (as Feynman put it) "swallow the doctor". The idea was incorporated into Feynman's 1959 essay There's Plenty of Room at the Bottom.

Digital immortality

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Digital_immortality 

 

Digital immortality (or "virtual immortality") is the hypothetical concept of storing (or transferring) a person's personality in more durable media, i.e., a computer. The result might look like an avatar behaving, reacting, and thinking like a person on the basis of that person's digital archive. After the death of the individual, this avatar could remain static or continue to learn and develop autonomously.

A considerable portion of transhumanists and singularitarians place great hope into the belief that they may eventually become immortal by creating one or many non-biological functional copies of their brains, thereby leaving their "biological shell". These copies may then "live eternally" in a version of digital "heaven" or paradise.

The realism of the concept

The National Science Foundation has awarded a half-million-dollar grant to the universities of Central Florida at Orlando and Illinois at Chicago to explore how researchers might use artificial intelligence, archiving, and computer imaging to create convincing, digital versions of real people, a possible first step toward virtual immortality.

The Digital Immortality Institute explores three factors necessary for digital immortality. First, at whatever level of implementation, avatars require guaranteed Internet accessibility. Next, avatars must be what users specify, and they must remain so. Finally, future representations must be secured before the living users are no more.

The aim of Dmitry Itskov's 2045 Initiative is to "create technologies enabling the transfer of an individual’s personality to a non-biological carrier, and extending existence, including to the point of immortality".

Method

Reaching digital immortality is a two-step process:

1. archiving and digitizing people,
2. making the avatar live

Archiving and digitizing people

According to Gordon Bell and Jim Gray from Microsoft Research, retaining every conversation that a person has ever heard is already realistic: it needs less than a terabyte of storage (for adequate quality). The speech or text recognition technologies are one of the biggest challenges of the concept. 

A second possibility would be to archive and analyze social Internet use to map the personality of people. By analyzing social Internet use during 50 years, it would be possible to model a society's culture, a society's way of thinking, and a society's interests.

Rothblatt envisions the creation of "mindfiles" – collections of data from all kinds of sources, including the photos we upload to Facebook, the discussions and opinions we share on forums or blogs, and other social media interactions that reflect our life experiences and our unique self.

Richard Grandmorin summarized the concept of digital immortality by the following equation: "semantic analysis + social internet use + Artificial Intelligence = immortality". 

Some find that photos, videos, soundclips, social media posts and other data of oneself could already be regarded as such an archiving.

Susanne Asche states:

As a hopefully minimalistic definition then, digital immortality can be roughly considered as involving a person-centric repository containing a copy of everything that a person sees, hears, says, or engenders over his or her lifespan, including photographs, videos, audio recordings, movies, television shows, music albums/CDs, newspapers, documents, diaries and journals, interviews, meetings, love letters, notes, papers, art pieces, and so on, and so on; and if not everything, then at least as much as the person has and takes the time and trouble to include. The person’s personality, emotion profiles, thoughts, beliefs, and appearance are also captured and integrated into an artificially intelligent, interactive, con-versational agent/avatar. This avatar is placed in charge of (and perhaps "equated" with) the collected material in the repository so that the agent can present the illusion of having the factual memories, thoughts, and beliefs of the person him/herself.

— Susanne Asche, Kulturelles Gedächtnis im 21. Jahrhundert: Tagungsband des internationalen Symposiums, Digital Immortality & Runaway Technology

Making the avatar alive

Defining the avatar to be alive allows it to communicate with the future in the sense that it continues to learn, evolve and interact with people, if they still exist. Technically, the operation exists to implement an artificial intelligence system to the avatar. This artificial intelligence system is then assumed to think and will react on the base of the archive. 

Rothblatt proposes the term "mindware" for software that is being developed with the goal of generating conscious AIs. Such software would read a person's "mindfile" to generate a "mindclone." Rothblatt also proposes a certain level of governmental approval for mindware, like an FDA certification, to ensure that the resulting mindclones are well made.

Calibration process

During the calibration process, the biological people are living at the same time as their artifact in silicon. The artifact in silicon is calibrated to be as close as possible to the person in question. During this process ongoing updates, synchronization, and interaction between the two minds would maintain the twin minds as one.

In fiction

• In the TV series Caprica a digital copy of a person is created and outlives its real counterpart after the person dies in a terrorist attack.
• In Greg Egan's Permutation City people can achieve quasi digital immortality by mind uploading a digital copy of themselves into a simulated reality.
• Memories with Maya is a novel on the concept of digital immortality.
• The Silicon Man describes Cryonics as a precursor to digital immortality.
• In the 1998 novel Vast by Linda Nagata "ghosts" are recorded memories and personalities that can be transferred to another body or kept in electronic storage, granting a limited form of immortality.
• In the TV series Captain Power and the Soldiers of the Future, Overmind and Lord Dread planned to digitize all human beings to be able to create a new world.
• In the TV series Black Mirror it commonly features the themes and ethics of digital humans, called "cookies," across multiple episodes. In San Junipero, for example, people's consciences are uploaded to the cloud.
• In the novel / Netflix series Altered Carbon, a person's memories and consciousness can be stored in a disk-shaped device called a cortical stack, which is implanted into the cervical vertebrae.
• In Frictional Games' SOMA, the story revolves around the problem of existing as a digital personality scan taken from a physical person.

How to Be an Antiracist

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/How_to_Be_an_Antiracist 

 

How to Be an Antiracist

Front cover
Author	Ibram X. Kendi
Subject	Civil rights
Publisher	Random House
Publication date	August 13, 2019
Pages	320
ISBN	9780525509288

How to Be an Antiracist is a 2019 non-fiction book by American author and historian Ibram X. Kendi. The book discusses concepts of racism and Kendi's proposals for anti-racist individual actions and systemic changes. It received positive critical reception. 

Background

At the time of authorship, Ibram X. Kendi was an assistant professor of African-American History at the University of Florida. He previously worked at the American University, where he founded the Antiracist Research and Policy Center. He wrote a 2016 book titled Stamped from the Beginning, about the origins of racism in America.

Synopsis

Kendi describes concepts of racism such as scientific racism, colorism and their intersection with demographics including gender, class and sexuality. He summarizes historical eras such as the scientific proposals of polygenism in Europe in the 1600s and racial segregation in the United States. The book also covers contemporary history such as the O. J. Simpson robbery case and 2000 United States presidential election. He also details experiences from his own life, including his change in beliefs over time, and observations from classes he has taught. Kendi comments on internalized racism and disputes the prejudice plus power model of racism. He suggests models for anti-racist individual actions and systemic changes.

Reception

The book was published in August 2019 to mixed, but generally positive reviews. In June 2020, following protests in the wake of the killing of George Floyd, sales of How to Be an Antiracist surged. The book was listed eighth and fifth in Publishers Weekly's hardcover non-fiction list on May 30 and June 6, respectively. It was listed third in USA Today's Best-Selling Books List of June 10. The book topped The New York Times Bestseller List in Hardcover Nonfiction list for sales in the week ending June 6. It has spent a total of 18 weeks on the list, as of the July 19 edition of the list.

Critical reception

Ayesha Pande praised the book in a starred review for Publishers Weekly, describing the prose as "thoughtful, sincere and polished" and the ideas as "boldly articulated" and "historically informed". Pande summarized, "This powerful book will spark many conversations". A starred review for Kirkus Reviews found it to be "not an easy read but an essential one". Jeffrey C. Stewart of The New York Times lauded it as the "most courageous book to date on the problem of race in the Western mind". Ericka Taylor of NPR praised the book as "clear and compelling", saying that it is "accessible" and "exemplifies a commitment to clarity".

In a mostly positive review by The Guardian's Afua Hirsch, the author received praise for "honesty in linking his personal struggles" to the book's subject, which Hirsch described as "brilliantly simple" and "dogmatic", but criticism for personal anecdotes that seem incomplete and for a style resembling a textbook too much. It was the Book of the Day in a review for The Observer in which Colin Grant found that the book "encourages self-reflection" and praised the writing style as "calm" but "insightful".

Coleman Hughes critiqued the book as "poorly argued, sloppily researched, insufficiently fact-checked, and occasionally self-contradictory". In the Washington Post, Randall Kennedy, praised Kendi's book for its candor, independence, and self-criticalness, but also critiqued it as having major flaws—especially being internally contradictory and poorly reasoned. Andrew Sullivan criticized the book as having the character of religious tract with overly simplistic distinctions between good and evil that cannot be falsified, and being sparse on practical suggestions.

Set (mathematics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Set_(mathematics) 

A set of polygons in an Euler diagram

 

In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. The arrangement of the objects in the set does not matter. For example, the numbers 2, 4, and 6 are distinct objects when considered separately, but when they are considered collectively they form a single set of size three, written as {2, 4, 6}, which could also be written as {2, 6, 4}.

The concept of a set is one of the most fundamental in mathematics. Developed at the end of the 19th century, set theory is now a ubiquitous part of mathematics, and can be used as a foundation from which nearly all of mathematics can be derived.

Etymology

The German word Menge, rendered as "set" in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite.

Definition

Passage with a translation of the original set definition of Georg Cantor. The German word Menge for set is translated with aggregate here.

 

A set is a well-defined collection of distinct objects. The objects that make up a set (also known as the set's elements or members) can be anything: numbers, people, letters of the alphabet, other sets, and so on. Georg Cantor, one of the founders of set theory, gave the following definition of a set at the beginning of his Beiträge zur Begründung der transfiniten Mengenlehre:

A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements.

For technical reasons, Cantor's definition turned out to be inadequate; today, in contexts where more rigor is required, one can use axiomatic set theory, in which the notion of a "set" is taken as a primitive notion and the properties of sets are defined by a collection of axioms. The most basic properties are that a set can have elements, and that two sets are equal (one and the same) if and only if every element of each set is an element of the other; this property is called the extensionality of sets.

Set notation

Main article: Set notation

There are two common ways of describing, or specifying the members of, a set: roster notation and set builder notation. These are examples of extensional and intensional definitions of sets, respectively.

Roster notation

The Roster notation (or enumeration notation) method of defining a set consist of listing each member of the set. More specifically, in roster notation (an example of extensional definition), the set is denoted by enclosing the list of members in curly brackets:

A = {4, 2, 1, 3}

B = {blue, white, red}.

For sets with many elements, the enumeration of members can be abbreviated. For instance, the set of the first thousand positive integers may be specified in roster notation as

{1, 2, 3, ..., 1000},

where the ellipsis ("...") indicates that the list continues in according to the demonstrated pattern.

In roster notation, listing a member repeatedly does not change the set, for example, the set {11, 6, 6} is identical to the set {11, 6}. Moreover, the order in which the elements of a set are listed is irrelevant (unlike for a sequence or tuple), so {6, 11} is yet again the same set.

Set-builder notation

In set-builder notation, the set is specified as a subset of a larger set, where the subset is determined by a statement or condition involving the elements. For example, a set F can be specified as follows:

${\displaystyle F=\{n\mid n{\text{ is an integer, and }}0\leq n\leq 19\}.}$

In this notation, the vertical bar ("|") means "such that", and the description can be interpreted as "F is the set of all numbers n, such that n is an integer in the range from 0 to 19 inclusive". Sometimes the colon (":") is used instead of the vertical bar.

Set-builder notation is an example of intensional definition.

Other ways of defining sets

Another method is by using a rule or semantic description:

A is the set whose members are the first four positive integers.

B is the set of colors of the French flag.

This is another example of intensional definition.

Membership

If B is a set and x is one of the objects of B, this is denoted as x ∈ B, and is read as "x is an element of B", as "x belongs to B", or "x is in B". If y is not a member of B then this is written as y ∉ B, read as "y is not an element of B", or "y is not in B".

For example, with respect to the sets A = {1, 2, 3, 4}, B = {blue, white, red}, and F = {n | n is an integer, and 0 ≤ n ≤ 19},

4 ∈ A and 12 ∈ F; and

20 ∉ F and green ∉ B.

Subsets

If every element of set A is also in B, then A is said to be a subset of B, written A ⊆ B (pronounced A is contained in B). Equivalently, one can write B ⊇ A, read as B is a superset of A, B includes A, or B contains A. The relationship between sets established by ⊆ is called inclusion or containment. Two sets are equal if they contain each other: A ⊆ B and B ⊆ A is equivalent to A = B.

If A is a subset of B, but not equal to B, then A is called a proper subset of B, written A ⊊ B, or simply A ⊂ B (A is a proper subset of B), or B ⊋ A (B is a proper superset of A, B ⊃ A).

The expressions A ⊂ B and B ⊃ A are used differently by different authors; some authors use them to mean the same as A ⊆ B (respectively B ⊇ A), whereas others use them to mean the same as A ⊊ B (respectively B ⊋ A). 

A is a subset of B

Examples:

• The set of all humans is a proper subset of the set of all mammals.
• {1, 3} ⊆ {1, 2, 3, 4}.
• {1, 2, 3, 4} ⊆ {1, 2, 3, 4}.

There is a unique set with no members, called the empty set (or the null set), which is denoted by the symbol ∅ (other notations are used; see empty set). The empty set is a subset of every set, and every set is a subset of itself:

• ∅ ⊆ A.
• A ⊆ A.

Partitions

A partition of a set S is a set of nonempty subsets of S such that every element x in S is in exactly one of these subsets. That is, the subsets are pairwise disjoint (meaning any two sets of the partition contain no element in common), and the union of all the subsets of the partition is S.

Power sets

The power set of a set S is the set of all subsets of S. The power set contains S itself and the empty set because these are both subsets of S. For example, the power set of the set {1, 2, 3} is {{1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, ∅}. The power set of a set S is usually written as P(S).

The power set of a finite set with n elements has 2ⁿ elements. For example, the set {1, 2, 3} contains three elements, and the power set shown above contains 2³ = 8 elements.

The power set of an infinite (either countable or uncountable) set is always uncountable. Moreover, the power set of a set is always strictly "bigger" than the original set in the sense that there is no way to pair every element of S with exactly one element of P(S). (There is never an onto map or surjection from S onto P(S).)

Cardinality

The cardinality of a set S, denoted |S|, is the number of members of S. For example, if B = {blue, white, red}, then |B| = 3. Repeated members in roster notation are not counted, so |{blue, white, red, blue, white}| = 3, too. 

The cardinality of the empty set is zero.

Some sets have infinite cardinality. The set N of natural numbers, for instance, is infinite. Some infinite cardinalities are greater than others. For instance, the set of real numbers has greater cardinality than the set of natural numbers. However, it can be shown that the cardinality of (which is to say, the number of points on) a straight line is the same as the cardinality of any segment of that line, of the entire plane, and indeed of any finite-dimensional Euclidean space.

Special sets

The natural numbers ℕ are contained in the integers ℤ, which are contained in the rational numbers ℚ, which are contained in the real numbers ℝ, which are contained in the complex numbers ℂ

There are some sets or kinds of sets that hold great mathematical importance and are referred to with such regularity that they have acquired special names and notational conventions to identify them. One of these is the empty set, denoted { } or ∅. A set with exactly one element, x, is a unit set, or singleton, {x}.

Many of these sets are represented using bold (e.g. P) or blackboard bold (e.g. ℙ) typeface.

Special sets of numbers include

• P or ℙ, denoting the set of all primes: P = {2, 3, 5, 7, 11, 13, 17, ...}.
• N or ${\displaystyle \mathbb {N} }$, denoting the set of all natural numbers: N = {0, 1, 2, 3, ...} (sometimes defined excluding 0).
• Z or ${\displaystyle \mathbb {Z} }$, denoting the set of all integers (whether positive, negative or zero): Z = {..., −2, −1, 0, 1, 2, ...}.
• Q or ℚ, denoting the set of all rational numbers (that is, the set of all proper and improper fractions): Q = {a/b | a, b ∈ Z, b ≠ 0}. For example, 1/4 ∈ Q and 11/6 ∈ Q. All integers are in this set since every integer a can be expressed as the fraction a/1 (Z ⊊ Q).
• R or ${\displaystyle \mathbb {R} }$, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, algebraic numbers that cannot be rewritten as fractions such as √2, as well as transcendental numbers such as π, e).
• C or ℂ, denoting the set of all complex numbers: C = {a + bi | a, b ∈ R}. For example, 1 + 2i ∈ C.
• H or ℍ, denoting the set of all quaternions: H = {a + bi + cj + dk | a, b, c, d ∈ R}. For example, 1 + i + 2j − k ∈ H.

Each of the above sets of numbers has an infinite number of elements, and each can be considered to be a proper subset of the sets listed below it. The primes are used less frequently than the others outside of number theory and related fields. 

Positive and negative sets are sometimes denoted by superscript plus and minus signs, respectively. For example, ℚ⁺ represents the set of positive rational numbers.

Basic operations

There are several fundamental operations for constructing new sets from given sets.

Unions

The union of A and B, denoted A ∪ B

Two sets can be "added" together. The union of A and B, denoted by A ∪ B, is the set of all things that are members of either A or B. 

Examples:

• {1, 2} ∪ {1, 2} = {1, 2}.
• {1, 2} ∪ {2, 3} = {1, 2, 3}.
• {1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}

Some basic properties of unions:

• A ∪ B = B ∪ A.
• A ∪ (B ∪ C) = (A ∪ B) ∪ C.
• A ⊆ (A ∪ B).
• A ∪ A = A.
• A ∪ ∅ = A.
• A ⊆ B if and only if A ∪ B = B.

Intersections

A new set can also be constructed by determining which members two sets have "in common". The intersection of A and B, denoted by A ∩ B, is the set of all things that are members of both A and B. If A ∩ B = ∅, then A and B are said to be disjoint. 

The intersection of A and B, denoted A ∩ B.

Examples:

• {1, 2} ∩ {1, 2} = {1, 2}.
• {1, 2} ∩ {2, 3} = {2}.
• {1, 2} ∩ {3, 4} = ∅.

Some basic properties of intersections:

• A ∩ B = B ∩ A.
• A ∩ (B ∩ C) = (A ∩ B) ∩ C.
• A ∩ B ⊆ A.
• A ∩ A = A.
• A ∩ ∅ = ∅.
• A ⊆ B if and only if A ∩ B = A.

Complements

The relative complement
of B in A

 

The complement of A in U

 

The symmetric difference of A and B

Two sets can also be "subtracted". The relative complement of B in A (also called the set-theoretic difference of A and B), denoted by A \ B (or A − B), is the set of all elements that are members of A but not members of B. It is valid to "subtract" members of a set that are not in the set, such as removing the element green from the set {1, 2, 3}; doing so has no effect. 

In certain settings all sets under discussion are considered to be subsets of a given universal set U. In such cases, U \ A is called the absolute complement or simply complement of A, and is denoted by A′.

• A′ = U \ A

Examples:

• {1, 2} \ {1, 2} = ∅.
• {1, 2, 3, 4} \ {1, 3} = {2, 4}.
• If U is the set of integers, E is the set of even integers, and O is the set of odd integers, then U \ E = E′ = O.

Some basic properties of complements:

• A \ B ≠ B \ A for A ≠ B.
• A ∪ A′ = U.
• A ∩ A′ = ∅.
• (A′)′ = A.
• ∅ \ A = ∅.
• A \ ∅ = A.
• A \ A = ∅.
• A \ U = ∅.
• A \ A′ = A and A′ \ A = A′.
• U′ = ∅ and ∅′ = U.
• A \ B = A ∩ B′.
• if A ⊆ B then A \ B = ∅.

An extension of the complement is the symmetric difference, defined for sets A, B as

${\displaystyle A\,\Delta \,B=(A\setminus B)\cup (B\setminus A).}$

For example, the symmetric difference of {7, 8, 9, 10} and {9, 10, 11, 12} is the set {7, 8, 11, 12}. The power set of any set becomes a Boolean ring with symmetric difference as the addition of the ring (with the empty set as neutral element) and intersection as the multiplication of the ring.

Cartesian product

A new set can be constructed by associating every element of one set with every element of another set. The Cartesian product of two sets A and B, denoted by A × B is the set of all ordered pairs (a, b) such that a is a member of A and b is a member of B.

Examples:

• {1, 2} × {red, white, green} = {(1, red), (1, white), (1, green), (2, red), (2, white), (2, green)}.
• {1, 2} × {1, 2} = {(1, 1), (1, 2), (2, 1), (2, 2)}.
• {a, b, c} × {d, e, f} = {(a, d), (a, e), (a, f), (b, d), (b, e), (b, f), (c, d), (c, e), (c, f)}.

Some basic properties of Cartesian products:

• A × ∅ = ∅.
• A × (B ∪ C) = (A × B) ∪ (A × C).
• (A ∪ B) × C = (A × C) ∪ (B × C).

Let A and B be finite sets; then the cardinality of the Cartesian product is the product of the cardinalities:

• | A × B | = | B × A | = | A | × | B |.

Applications

Set theory is seen as the foundation from which virtually all of mathematics can be derived. For example, structures in abstract algebra, such as groups, fields and rings, are sets closed under one or more operations. 

One of the main applications of naive set theory is constructing relations. A relation from a domain A to a codomain B is a subset of the Cartesian product A × B. For example, considering the set S = { rock, paper, scissors } of shapes in the game of the same name, the relation "beats" from S to S is the set B = { (scissors,paper), (paper,rock), (rock,scissors) }; thus x beats y in the game if the pair (x,y) is a member of B. Another example is the set F of all pairs (x, x²), where x is real. This relation is a subset of R' × R, because the set of all squares is subset of the set of all real numbers. Since for every x in R, one, and only one, pair (x,...) is found in F, it is called a function. In functional notation, this relation can be written as F(x) = x².

Axiomatic set theory

Although initially naive set theory, which defines a set merely as any well-defined collection, was well accepted, it soon ran into several obstacles. It was found that this definition spawned several paradoxes, most notably:

• Russell's paradox – It shows that the "set of all sets that do not contain themselves," i.e. the "set" {x|x is a set and x ∉ x} does not exist.
• Cantor's paradox – It shows that "the set of all sets" cannot exist.

The reason is that the phrase well-defined is not very well-defined. It was important to free set theory of these paradoxes because nearly all of mathematics was being redefined in terms of set theory. In an attempt to avoid these paradoxes, set theory was axiomatized based on first-order logic, and thus axiomatic set theory was born. 

For most purposes, however, naive set theory is still useful.

Principle of inclusion and exclusion

The inclusion-exclusion principle can be used to calculate the size of the union of sets: the size of the union is the size of the two sets, minus the size of their intersection.

 

The inclusion–exclusion principle is a counting technique that can be used to count the number of elements in a union of two sets, if the size of each set and the size of their intersection are known. It can be expressed symbolically as

${\displaystyle |A\cup B|=|A|+|B|-|A\cap B|.}$

A more general form of the principle can be used to find the cardinality of any finite union of sets:

${\displaystyle {\begin{aligned}\left|A_{1}\cup A_{2}\cup A_{3}\cup \ldots \cup A_{n}\right|=&\left(\left|A_{1}\right|+\left|A_{2}\right|+\left|A_{3}\right|+\ldots \left|A_{n}\right|\right)\\&{}-\left(\left|A_{1}\cap A_{2}\right|+\left|A_{1}\cap A_{3}\right|+\ldots \left|A_{n-1}\cap A_{n}\right|\right)\\&{}+\ldots \\&{}+\left(-1\right)^{n-1}\left(\left|A_{1}\cap A_{2}\cap A_{3}\cap \ldots \cap A_{n}\right|\right).\end{aligned}}}$

De Morgan's laws

Augustus De Morgan stated two laws about sets. 

If A and B are any two sets then,

• (A ∪ B)′ = A′ ∩ B′

The complement of A union B equals the complement of A intersected with the complement of B.

• (A ∩ B)′ = A′ ∪ B′

The complement of A intersected with B is equal to the complement of A union to the complement of B.