In mathematics, the **tombstone**, **halmos**, **end-of-proof**, or **Q.E.D.** symbol "∎" (or "□") is a symbol used to denote the end of a proof, in place of the traditional abbreviation "Q.E.D." for the Latin phrase "* quod erat demonstrandum *", meaning "which was to be demonstrated".^{ [1] } In magazines, it is one of the various symbols used to indicate the end of an article.^{ [2] }

In Unicode, it is represented as character U+220E∎END OF PROOF (HTML `∎`

). Its graphic form varies, as it may be a hollow or filled rectangle or square.

In AMS-LaTeX, the symbol is automatically appended at the end of a proof environment `\begin{proof}` ... `\end{proof}`. It can also be obtained from the commands `\qedsymbol`, `\qedhere`

or `\qed` (the latter causes the symbol to be right aligned).^{ [3] }

It is sometimes called a "Halmos finality symbol" or "halmos" after the mathematician Paul Halmos, who first used it in a mathematical context in 1950.^{ [4] } He got the idea of using it from seeing it was being used to indicate the end of articles in magazines. In his memoir *I Want to Be a Mathematician*, he wrote the following:^{ [5] }

The symbol is definitely not my invention — it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like ▯, and is used to indicate an end, usually the end of a proof. It is most frequently called the 'tombstone', but at least one generous author referred to it as the 'halmos'.

- ↑ "The Definitive Glossary of Higher Mathematical Jargon — Q.E.D."
*Math Vault*. 2019-08-01. Retrieved 2019-11-04. - ↑ Foster, A. J. "Tombstones in Typography | AJ Foster".
*aj-foster.com*. Retrieved 2019-11-05. - ↑ "LaTeX/Theorems - Wikibooks, open books for an open world".
*en.wikibooks.org*. Retrieved 2019-11-05. - ↑ Halmos, Paul R. (Paul Richard), 1916-2006. (1950).
*Measure theory*. New York: Van Nostrand. pp. 6. ISBN 0387900888. OCLC 529634.CS1 maint: multiple names: authors list (link) - ↑ Paul R. Halmos,
*I Want to Be a Mathematician: An Automathography*, 1985, p. 403.

**Naive set theory** is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naive set theory is defined informally, in natural language. It describes the aspects of mathematical sets familiar in discrete mathematics, and suffices for the everyday use of set theory concepts in contemporary mathematics.

**Epsilon** is the fifth letter of the Greek alphabet, corresponding phonetically to a mid front unrounded vowel. In the system of Greek numerals it also has the value five. It was derived from the Phoenician letter He . Letters that arose from epsilon include the Roman E, Ë and Ɛ, and Cyrillic Е, È, Ё, Є and Э.

In logic and related fields such as mathematics and philosophy, **if and only if** is a biconditional logical connective between statements, where either both statements are true or both are false.

**Mathematics** includes the study of such topics as quantity, structure (algebra), space (geometry), and change. It has no generally accepted definition.

A **number** is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can be represented by symbols, called *numerals*; for example, "5" is a numeral that represents the number five. As only a relatively small number of symbols can be memorized, basic numerals are commonly organized in a numeral system, which is an organized way to represent any number. The most common numeral system is the Hindu–Arabic numeral system, which allows for the representation of any number using a combination of ten fundamental numeric symbols, called digits. In addition to their use in counting and measuring, numerals are often used for labels, for ordering, and for codes. In common usage, a *numeral* is not clearly distinguished from the *number* that it represents.

The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. **Nonstandard analysis** instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers.

**Q.E.D.** or **QED** is an initialism of the Latin phrase "* quod erat demonstrandum*", literally meaning "what was to be shown". Traditionally, the abbreviation is placed at the end of a mathematical proof or philosophical argument in print publications to indicate that the proof or the argument is complete, and hence is used with the meaning "thus it has been demonstrated".

In logic and mathematics, **proof by contradiction** is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as **indirect proof**, **proof by assuming the opposite**, and * reductio ad impossibile*.

In mathematics, a **set** is a well-defined collection of distinct elements or members. The elements that make up a set can be anything: people, letters of the alphabet, or mathematical objects, such as numbers, points in space, lines or other geometrical shapes, algebraic constants and variables, or other sets. Two sets are equal if and only if they have precisely the same elements.

In mathematics, a **theorem** is a non-self-evident statement that has been proven to be true, either on the basis of generally accepted statements such as axioms or on the basis of previously established statements such as other theorems. A theorem is hence a logical consequence of the axioms, with a proof of the theorem being a logical argument which establishes its truth through the inference rules of a deductive system. As a result, the proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally *deductive*, in contrast to the notion of a scientific law, which is *experimental*.

**Blackboard bold** is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets. One way of producing blackboard bold is to double-strike a character with a small offset on a typewriter. Thus they are also referred to as **double struck**.

A **mathematical symbol** is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula. As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics.

A **mathematical proof** is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in *all* possible cases. An unproven proposition that is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work.

**Paul Richard Halmos** was a Hungarian-born American mathematician and statistician who made fundamental advances in the areas of mathematical logic, probability theory, statistics, operator theory, ergodic theory, and functional analysis. He was also recognized as a great mathematical expositor. He has been described as one of The Martians.

The **language of mathematics** is the system used by mathematicians to communicate mathematical ideas among themselves, and is distinct from natural languages in that it aims to communicate abstract, logical ideas with precision and unambiguity.

The **equal sign** or **equality sign**, **=**, is a mathematical symbol used to indicate equality in some well-defined sense. It was invented in 1557 by Robert Recorde. In an equation, the equal sign is placed between two expressions that have the same value, or for which one studies the conditions under which they have the same value. In Unicode and ASCII, it has the code point 3D.

The **dangerous bend** or **caution** symbol **☡** was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group. It resembles a road sign that indicates a "dangerous bend" in the road ahead, and is used to mark passages tricky on a first reading or with an especially difficult argument.

In mathematical logic and computer science the symbol has taken the name **turnstile** because of its resemblance to a typical turnstile if viewed from above. It is also referred to as **tee** and is often read as "yields", "proves", "satisfies" or "entails".

The following table lists many specialized symbols commonly used in mathematics, ordered by their introduction date. Note that the table can also be ordered alphabetically by clicking on the relevant header title.

- Miller, Jeff (September 29, 2007),
*Earliest Uses of Symbols of Set Theory and Logic*, retrieved June 26, 2010

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