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75-427: A formula such as x = y , {\displaystyle x=y,} where x and y are any expressions, means that x and y denote or represent the same object. For example, are two notations for the same number. Similarly, using set builder notation , since the two sets have the same elements. (This equality results from the axiom of extensionality that is often expressed as "two sets that have

150-409: A BOOLEAN column directly as a predicate in a WHERE clause. In MySQL , BOOLEAN is treated as an alias of TINYINT ( 1 ) ; TRUE is the same as integer 1 and FALSE is the same is integer 0. Any non-zero integer is true in conditions. Tableau Software has a BOOLEAN data type. The literal of a Boolean value is True or False . The Tableau INT() function converts

225-543: A and b , if a = b , then a ≥ 0 implies b ≥ 0 (here, ϕ ( x ) {\displaystyle \phi (x)} is x ≥ 0 ) These properties offer a formal reinterpretation of equality from how it is defined in standard Zermelo–Fraenkel set theory (ZFC) or other formal foundations . In ZFC, equality only means that two sets have the same elements. However, outside of set theory , mathematicians don't tend to view their objects of interest as sets. For instance, many mathematicians would say that

300-578: A complete axiomatization of equality, meaning, if they were to define equality, then the converse of the second statement must be true. The converse of the Substitution property is the identity of indiscernibles , which states that two distinct things cannot have all their properties in common. In mathematics, the identity of indiscernibles is usually rejected since indiscernibles in mathematical logic are not necessarily forbidden. Set equality in ZFC

375-417: A congruence relation . In logic , a predicate is a proposition which may have some free variables . Equality is a predicate, which may be true for some values of the variables (if any) and false for other values. More specifically, equality is a binary relation (i.e., a two-argument predicate) which may produce a truth value ( true or false ) from its arguments. In computer programming , equality

450-432: A Boolean to a number, returning 1 for True and 0 for False. Forth (programming language) has no Boolean type, it uses regular integers: value 0 (all bits low) represents false, and -1 (all bits high) represents true. This allows the language to define only one set of logical operators, instead of one for mathematical calculations and one for conditions. In some programming languages, any expression can be evaluated in

525-411: A Boolean variable. C++ has a separate Boolean data type bool , but with automatic conversions from scalar and pointer values that are very similar to those of C. This approach was adopted also by many later languages, especially by some scripting languages such as AWK . The D programming language has a proper Boolean data type bool . The bool type is a byte-sized type that can only hold

600-604: A SQL BOOLEAN behaves like Booleans in other languages, which can store only TRUE and FALSE values. However, if it is nullable, which is the default like all other SQL data types, it can have the special null value also. Although the SQL standard defines three literals for the BOOLEAN type – TRUE, FALSE, and UNKNOWN — it also says that the NULL BOOLEAN and UNKNOWN "may be used interchangeably to mean exactly

675-425: A choice – any statement that identifies them "depends on choice of identification". This distinction, between equality and isomorphism , is of fundamental importance in category theory and is one motivation for the development of category theory. In some cases, one may consider as equal two mathematical objects that are only equivalent for the properties and structure being considered. The word congruence (and

750-572: A comma instead of the symbol ∧ {\displaystyle \land } . In general, it is not a good idea to consider sets without defining a domain of discourse , as this would represent the subset of all possible things that may exist for which the predicate is true. This can easily lead to contradictions and paradoxes. For example, Russell's paradox shows that the expression { x   |   x ∉ x } , {\displaystyle \{x~|~x\not \in x\},} although seemingly well formed as

825-476: A complete axiomatization. However, apart from cases dealing with indiscernibles, these properties taken as axioms of equality are equivalent to equality as defined in ZFC. These are sometimes taken as the definition of equality, such as in some areas of first-order logic . The Law of identity is distinct from reflexivity in two main ways: first, the Law of Identity applies only to cases of equality, and second, it

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900-449: A context that expects a Boolean data type. Typically (though this varies by programming language) expressions like the number zero , the empty string , empty lists, and null are treated as false, and strings with content (like "abc"), other numbers, and objects evaluate to true. Sometimes these classes of expressions are called falsy and truthy. For example, in Lisp , nil , the empty list,

975-408: A list of the yielded variables using the "yield" keyword. Consider these set-builder notation examples in some programming languages: The set builder notation and list comprehension notation are both instances of a more general notation known as monad comprehensions , which permits map/filter-like operations over any monad with a zero element . Boolean data type In computer science ,

1050-442: A numeric value of zero (integer or fractional), the null value ( None ), the empty string, and empty containers (lists, sets , etc.) are considered Boolean false; all other values are considered Boolean true by default. Classes can define how their instances are treated in a Boolean context through the special method __nonzero__ (Python 2) or __bool__ (Python 3). For containers, __len__ (the special method for determining

1125-965: A programmer-specified Boolean condition evaluates to true or false. It is a special case of a more general logical data type— logic does not always need to be Boolean (see probabilistic logic ). In programming languages with a built-in Boolean data type, such as Pascal and Java , the comparison operators such as > and ≠ are usually defined to return a Boolean value. Conditional and iterative commands may be defined to test Boolean-valued expressions. Languages with no explicit Boolean data type, like C90 and Lisp , may still represent truth values by some other data type. Common Lisp uses an empty list for false, and any other value for true. The C programming language uses an integer type, where relational expressions like i > j and logical expressions connected by && and || are defined to have value 1 if true and 0 if false, whereas

1200-608: A separate Boolean data type BOOL , with possible values being YES or NO , equivalents of true and false respectively. Also, in Objective-C compilers that support C99, C's _Bool type can be used, since Objective-C is a superset of C. In Java , the value of the boolean data type can only be either true or false . Perl has no Boolean data type. Instead, any value can behave as Boolean in Boolean context (condition of if or while statement, argument of && or || , etc.). The number 0 ,

1275-467: A set builder expression, cannot define a set without producing a contradiction. In cases where the set E is clear from context, it may be not explicitly specified. It is common in the literature for an author to state the domain ahead of time, and then not specify it in the set-builder notation. For example, an author may say something such as, "Unless otherwise stated, variables are to be taken to be natural numbers," though in less formal contexts where

1350-406: A set. For example, the set of all solution pairs ( x , y ) {\displaystyle (x,y)} of the equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} forms the unit circle in analytic geometry ; therefore, this equation is called the equation of the unit circle . See also: Equation solving An identity

1425-443: A set: those binary relations that are reflexive , symmetric and transitive . The identity relation is an equivalence relation. Conversely, let R be an equivalence relation, and let us denote by x the equivalence class of x , consisting of all elements z such that x R z . Then the relation x R y is equivalent with the equality x  =  y . It follows that equality is the finest equivalence relation on any set S in

1500-398: A specific format descriptor (' L ') is provided for the parsing or formatting of logical values. The language Lisp (1958) never had a built-in Boolean data type. Instead, conditional constructs like cond assume that the logical value false is represented by the empty list () , which is defined to be the same as the special atom nil or NIL ; whereas any other s-expression

1575-459: A third state, called UNKNOWN , when comparison with NULL is made. The SQL92 standard introduced IS (NOT) TRUE, IS (NOT) FALSE, and IS (NOT) UNKNOWN operators which evaluate a predicate, which predated the introduction of Boolean type in SQL:1999 . The SQL:1999 standard introduced a BOOLEAN data type as an optional feature (T031). When restricted by a NOT NULL constraint,

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1650-434: A variable, a colon or vertical bar separator, and a predicate. Thus there is a variable on the left of the separator, and a rule on the right of it. These three parts are contained in curly brackets: or The vertical bar (or colon) is a separator that can be read as " such that ", "for which", or "with the property that". The formula Φ( x ) is said to be the rule or the predicate . All values of x for which

1725-406: Is 1 or true since the expression evaluates to 1. The above will render an error, as variable v cannot be evaluated as 0 or 1. Python , from version 2.3 forward, has a bool type which is a subclass of int , the standard integer type. It has two possible values: True and False , which are special versions of 1 and 0 respectively and behave as such in arithmetic contexts. Also,

1800-475: Is a mathematical notation for describing a set by stating the properties that its members must satisfy. Defining sets by properties is also known as set comprehension , set abstraction or as defining a set's intension . Set-builder notation can be used to describe a set that is defined by a predicate , that is, a logical formula that evaluates to true for an element of the set, and false otherwise. In this form, set-builder notation has three parts:

1875-617: Is a conjunction Φ 1 ( x ) ∧ Φ 2 ( x ) {\displaystyle \Phi _{1}(x)\land \Phi _{2}(x)} , then { x ∈ E ∣ Φ ( x ) } {\displaystyle \{x\in E\mid \Phi (x)\}} is sometimes written { x ∈ E ∣ Φ 1 ( x ) , Φ 2 ( x ) } {\displaystyle \{x\in E\mid \Phi _{1}(x),\Phi _{2}(x)\}} , using

1950-571: Is an enumerated type ) was adopted by most later languages which had enumerated types, such as Modula , Ada , and Haskell . Initial implementations of the language C (1972) provided no Boolean type, and to this day Boolean values are commonly represented by integers ( int s) in C programs. The comparison operators ( > , == , etc.) are defined to return a signed integer ( int ) result, either 0 (for false) or 1 (for true). Logical operators ( && , || , ! , etc.) and condition-testing statements ( if , while ) assume that zero

2025-445: Is an equality that is true for all values of its variables in a given domain. An "equation" may sometimes mean an identity, but more often than not, it specifies a subset of the variable space to be the subset where the equation is true. An example is ( x + 1 ) ( x + 1 ) = x 2 + 2 x + 1 {\displaystyle \left(x+1\right)\left(x+1\right)=x^{2}+2x+1}

2100-425: Is called a Boolean -valued expression , and its computation from the two expressions is known as comparison . See also: Relational operator § Equality An equation is the problem of finding values of some variable, called unknown , for which the specified equality is true. Each value of the unknown for which the equation holds is called a solution of the given equation; also stated as satisfying

2175-425: Is capable of declairing these indiscernibles as not equal, but an equality solely defined by these properties is not. Thus these properties form a strictly weaker notion of equality than set equality in ZFC. Outside of pure math , the identity of indiscernibles has attracted much controversy and criticism, especially from corpuscular philosophy and quantum mechanics . This is why the properties are said to not form

2250-500: Is commonly used as a workaround to store Boolean values, but workarounds need to be used such as UPDATE t SET flag = IIF ( col IS NOT NULL , 1 , 0 ) WHERE flag = 0 to convert between the integer and Boolean expression. Microsoft Access , which uses the Access Database Engine (ACE/JET), also does not have a Boolean data type. Similar to MS SQL Server, it uses a BIT data type. In Access it

2325-503: Is false and all other values are true. After enumerated types ( enum s) were added to the American National Standards Institute version of C, ANSI C (1989), many C programmers got used to defining their own Boolean types as such, for readability reasons. However, enumerated types are equivalent to integers according to the language standards; so the effective identity between Booleans and integers

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2400-440: Is intentional. This makes it an incomplete axiomatization of equality. That is, it does not say what equality is , only what "equality" must satify. However, the two axioms as stated are still generally useful, even as an incomplete axiomatization of equality, as they are usually sufficient for deducing most properties of equality that mathematicians care about. (See the following subsection) If these properties were to define

2475-564: Is interpreted as true . For convenience, most modern dialects of Lisp predefine the atom t to have value t , so that t can be used as a mnemonic notation for true . This approach ( any value can be used as a Boolean value ) was retained in most Lisp dialects ( Common Lisp , Scheme , Emacs Lisp ), and similar models were adopted by many scripting languages , even ones having a distinct Boolean type or Boolean values; although which values are interpreted as false and which are true vary from language to language. In Scheme, for example,

2550-482: Is known as a Yes/No data type which can have two values; Yes (True) or No (False). The BIT data type in Access can also can be represented numerically; True is −1 and False is 0. This differs to MS SQL Server in two ways, even though both are Microsoft products: PostgreSQL has a distinct BOOLEAN type as in the standard, which allows predicates to be stored directly into a BOOLEAN column, and allows using

2625-451: Is not restricted to elements of a set. However, many mathematicians refer to both as "Reflexivity", which is generally harmless. This is also sometimes included in the axioms of equality, but isn't necessary as it can be deduced from the other two axioms as shown above. There are some logic systems that do not have any notion of equality. This reflects the undecidability of the equality of two real numbers , defined by formulas involving

2700-525: Is not supported at all, neither as a standalone data type nor representable as an integer. It shows the error message "An expression of non-Boolean type specified in a context where a condition is expected" if a column is directly used in the WHERE clause, e.g. SELECT a FROM t WHERE a , while a statement such as SELECT column IS NOT NULL FROM t yields a syntax error. The BIT data type, which can only store integers 0 and 1 apart from NULL ,

2775-406: Is not transitive (since many small differences can add up to something big). However, equality almost everywhere is transitive. A questionable equality under test may be denoted using the = ? {\displaystyle {\stackrel {?}{=}}} symbol . Viewed as a relation , equality is the archetype of the more general concept of an equivalence relation on

2850-427: Is often described through the following properties: ∀ a ( a = a ) {\displaystyle \forall a(a=a)} ( a = b ) ⟹ [ ϕ ( a ) ⇒ ϕ ( b ) ] {\displaystyle (a=b)\implies {\bigl [}\phi (a)\Rightarrow \phi (b){\bigr ]}} For example: For all real numbers

2925-526: Is replaced with any number, then the two expressions take the same value. This may also be interpreted as saying that the two sides of the equals sign represent the same function (equality of functions), or that the two expressions denote the same polynomial (equality of polynomials). The word is derived from the Latin aequālis ("equal", "like", "comparable", "similar"), which itself stems from aequus ("equal", "level", "fair", "just"). If restricted to

3000-716: Is still valid for C programs. Standard C (since C99 ) provides a Boolean type, called _Bool . By including the header stdbool.h , one can use the more intuitive name bool and the constants true and false . The language guarantees that any two true values will compare equal (which was impossible to achieve before the introduction of the type). Boolean values still behave as integers, can be stored in integer variables, and used anywhere integers would be valid, including in indexing, arithmetic, parsing, and formatting. This approach ( Boolean values are just integers ) has been retained in all later versions of C. Note, that this does not mean that any integer value can be stored in

3075-643: Is true for all real numbers x {\displaystyle x} . There is no standard notation that distinguishes an equation from an identity, or other use of the equality relation: one has to guess an appropriate interpretation from the semantics of expressions and the context. Sometimes, but not always, an identity is written with a triple bar : ( x + 1 ) ( x + 1 ) ≡ x 2 + 2 x + 1. {\displaystyle \left(x+1\right)\left(x+1\right)\equiv x^{2}+2x+1.} In mathematical logic and mathematical philosophy , equality

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3150-536: The Boolean (sometimes shortened to Bool) is a data type that has one of two possible values (usually denoted true and false ) which is intended to represent the two truth values of logic and Boolean algebra . It is named after George Boole , who first defined an algebraic system of logic in the mid 19th century. The Boolean data type is primarily associated with conditional statements, which allow different actions by changing control flow depending on whether

3225-535: The false value is an atom distinct from the empty list, so the latter is interpreted as true . Common Lisp, on the other hand, also provides the dedicated boolean type, derived as a specialization of the symbol. The language Pascal (1970) popularized the concept of programmer-defined enumerated types , previously available with different nomenclature in COBOL , FACT and JOVIAL . A built-in Boolean data type

3300-417: The integers , the basic arithmetic operations , the logarithm and the exponential function . In other words, there cannot exist any algorithm for deciding such an equality (see Richardson's theorem ). The binary relation " is approximately equal " (denoted by the symbol ≈ {\displaystyle \approx } ) between real numbers or other things, even if more precisely defined,

3375-662: The strings "0" and "" , the empty list () , and the special value undef evaluate to false. All else evaluates to true. Lua has a Boolean data type, but non-Boolean values can also behave as Booleans. The non-value nil evaluates to false, whereas every other data type value evaluates to true. This includes the empty string "" and the number 0 , which are very often considered false in other languages. PL/I has no Boolean data type. Instead, comparison operators generate BIT(1) values; '0'B represents false and '1'B represents true . The operands of, e.g., & , | , ¬ , are converted to bit strings and

3450-425: The true and false values belong to separate classes , e.g., True and False , respectively, so there is no one Boolean type . In SQL , which uses a three-valued logic for explicit comparisons because of its special treatment of Nulls , the Boolean data type (introduced in SQL:1999 ) is also defined to include more than two truth values, so that SQL Booleans can store all logical values resulting from

3525-482: The associated symbol ≅ {\displaystyle \cong } ) is frequently used for this kind of equality, and is defined as the quotient set of the isomorphism classes between the objects. In geometry for instance, two geometric shapes are said to be equal or congruent when one may be moved to coincide with the other, and the equality/congruence relation is the isomorphism classes of isometries between shapes. Similarly to isomorphisms of sets,

3600-416: The axiom of extensionality states that two sets which contain the same elements are the same set. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by Lévy. In first-order logic without equality, two sets are defined to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets are contained in

3675-557: The conditional IF statement takes an arithmetic expression and branches to one of three locations according to its sign; see arithmetic IF . FORTRAN IV (1962), however, follows the ALGOL 60 example by providing a Boolean data type ( LOGICAL ), truth literals ( .TRUE. and .FALSE. ), logical IF statement, Boolean-valued numeric comparison operators ( .EQ. , .GT. , etc.), and logical operators ( .NOT. , .AND. , .OR. , .EQV. , and .NEQV. ). In FORMAT statements,

3750-454: The difference between isomorphisms and equality/congruence between such mathematical objects with properties and structure was one motivation for the development of category theory , as well as for homotopy type theory and univalent foundations . Equality of sets is axiomatized in set theory in two different ways, depending on whether the axioms are based on a first-order language with or without equality. In first-order logic with equality,

3825-753: The domain can be assumed, a written mention is often unnecessary. The following examples illustrate particular sets defined by set-builder notation via predicates. In each case, the domain is specified on the left side of the vertical bar, while the rule is specified on the right side. An extension of set-builder notation replaces the single variable x with an expression . So instead of { x ∣ Φ ( x ) } {\displaystyle \{x\mid \Phi (x)\}} , we may have { f ( x ) ∣ Φ ( x ) } , {\displaystyle \{f(x)\mid \Phi (x)\},} which should be read For example: When inverse functions can be explicitly stated,

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3900-401: The domain qualifiers. For example, because the two rule predicates are logically equivalent: This equivalence holds because, for any real number x , we have x 2 = 1 {\displaystyle x^{2}=1} if and only if x is a rational number with | x | = 1 {\displaystyle |x|=1} . In particular, both sets are equal to

3975-490: The elements of E that satisfy Φ : The set Y obtained from this axiom is exactly the set described in set builder notation as { x ∈ E ∣ Φ ( x ) } {\displaystyle \{x\in E\mid \Phi (x)\}} . A similar notation available in a number of programming languages (notably Python and Haskell ) is the list comprehension , which combines map and filter operations over one or more lists . In Python,

4050-450: The elements of a given set S {\displaystyle S} , those first three properties make equality an equivalence relation on S {\displaystyle S} . In fact, equality is the unique equivalence relation on S {\displaystyle S} whose equivalence classes are all singletons . Given operations over S {\displaystyle S} , that last property makes equality

4125-417: The equation. For example, the equation x 2 − 6 x + 5 = 0 {\displaystyle x^{2}-6x+5=0} has the values x = 1 {\displaystyle x=1} and x = 5 {\displaystyle x=5} as its only solutions. The terminology is used similarly for equations with several unknowns. An equation can be used to define

4200-726: The evaluation of predicates in SQL. A column of Boolean type can be restricted to just TRUE and FALSE though. One of the earliest programming languages to provide an explicit BOOLEAN data type is ALGOL 60 (1960) with values true and false and logical operators denoted by symbols ' ∧ {\displaystyle \wedge } ' (and), ' ∨ {\displaystyle \vee } ' (or), ' ⊃ {\displaystyle \supset } ' (implies), ' ≡ {\displaystyle \equiv } ' (equivalence), and ' ¬ {\displaystyle \neg } ' (not). Due to input device and character set limits on many computers of

4275-468: The expression " 1 ∪ 2 {\displaystyle 1\cup 2} " (see union ) is an abuse of notation or meaningless. This is a more abstracted framework which can be grounded in ZFC (that is, both axioms can be proved within ZFC as well as most other formal foundations), but is closer to how most mathematicians use equality. Note that this says "Equality implies these two properties" not that "These properties define equality"; this

4350-481: The expression on the left can be eliminated through simple substitution. Consider the example set { 2 t + 1 ∣ t ∈ Z } {\displaystyle \{2t+1\mid t\in \mathbb {Z} \}} . Make the substitution u = 2 t + 1 {\displaystyle u=2t+1} , which is to say t = ( u − 1 ) / 2 {\displaystyle t=(u-1)/2} , then replace t in

4425-498: The left of the vertical bar: or by adjoining it to the predicate: The ∈ symbol here denotes set membership , while the ∧ {\displaystyle \land } symbol denotes the logical "and" operator, known as logical conjunction . This notation represents the set of all values of x that belong to some given set E for which the predicate is true (see " Set existence axiom " below). If Φ ( x ) {\displaystyle \Phi (x)}

4500-543: The length of containers) is used if the explicit Boolean conversion method is not defined. In Ruby , in contrast, only nil (Ruby's null value) and a special false object are false ; all else (including the integer 0 and empty arrays) is true . Booleans appear in SQL when a condition is needed, such as WHERE clause, in form of predicate which is produced by using operators such as comparison operators, IN operator, IS (NOT) NULL etc. However, apart from TRUE and FALSE , these operators can also yield

4575-479: The operations are performed on each bit. The element-expression of an IF statement is true if any bit is 1. Rexx has no Boolean data type. Instead, comparison operators generate 0 or 1; 0 represents false and 1 represents true . The operands of, e.g., & , | , ¬ , must be 0 or 1. Tcl has no separate Boolean type. Like in C, the integers 0 (false) and 1 (true—in fact any nonzero integer) are used. Examples of coding: The above will show V

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4650-438: The predicate holds (is true) belong to the set being defined. All values of x for which the predicate does not hold do not belong to the set. Thus { x ∣ Φ ( x ) } {\displaystyle \{x\mid \Phi (x)\}} is the set of all values of x that satisfy the formula Φ . It may be the empty set , if no value of x satisfies the formula. A domain E can appear on

4725-479: The same elements are equal".) The truth of an equality depends on an interpretation of its members. In the above examples, the equalities are true if the members are interpreted as numbers or sets, but are false if the members are interpreted as expressions or sequences of symbols. An identity , such as ( x + 1 ) 2 = x 2 + 2 x + 1 , {\displaystyle (x+1)^{2}=x^{2}+2x+1,} means that if x

4800-478: The same rational number (the same point on a number line). This distinction gives rise to the notion of a quotient set . Similarly, the sets are not equal sets – the first consists of letters, while the second consists of numbers – but they are both sets of three elements and thus isomorphic, meaning that there is a bijection between them. For example However, there are other choices of isomorphism, such as and these sets cannot be identified without making such

4875-411: The same sets. Set builder notation { n ∣ ∃ k ∈ Z , n = 2 k } {\displaystyle \{n\mid \exists k\in \mathbb {Z} ,n=2k\}} The set of all even integers , expressed in set-builder notation. In set theory and its applications to logic , mathematics , and computer science , set-builder notation

4950-657: The same thing". This has caused some controversy because the identification subjects UNKNOWN to the equality comparison rules for NULL. More precisely UNKNOWN = UNKNOWN is not TRUE but UNKNOWN/NULL . As of 2012 few major SQL systems implement the T031 feature. Firebird and PostgreSQL are notable exceptions, although PostgreSQL implements no UNKNOWN literal; NULL can be used instead. The treatment of Boolean values differs between SQL systems. For example, in Microsoft SQL Server , Boolean value

5025-564: The sense that it is the relation that has the smallest equivalence classes (every class is reduced to a single element). In some contexts, equality is sharply distinguished from equivalence or isomorphism . For example, one may distinguish fractions from rational numbers , the latter being equivalence classes of fractions: the fractions 1 / 2 {\displaystyle 1/2} and 2 / 4 {\displaystyle 2/4} are distinct as fractions (as different strings of symbols) but they "represent"

5100-428: The set { − 1 , 1 } {\displaystyle \{-1,1\}} . In many formal set theories, such as Zermelo–Fraenkel set theory , set builder notation is not part of the formal syntax of the theory. Instead, there is a set existence axiom scheme , which states that if E is a set and Φ( x ) is a formula in the language of set theory, then there is a set Y whose members are exactly

5175-418: The set builder notation to find Two sets are equal if and only if they have the same elements. Sets defined by set builder notation are equal if and only if their set builder rules, including the domain specifiers, are equivalent. That is if and only if Therefore, in order to prove the equality of two sets defined by set builder notation, it suffices to prove the equivalence of their predicates, including

5250-465: The set-builder's braces are replaced with square brackets, parentheses, or curly braces, giving list, generator , and set objects, respectively. Python uses an English-based syntax. Haskell replaces the set-builder's braces with square brackets and uses symbols, including the standard set-builder vertical bar. The same can be achieved in Scala using Sequence Comprehensions, where the "for" keyword returns

5325-402: The test parts of if , while , for , etc., treat any non-zero value as true. Indeed, a Boolean variable may be regarded (and implemented) as a numerical variable with one binary digit ( bit ), or as a bit string of length one, which can store only two values. The implementation of Booleans in computers are most likely represented as a full word , rather than a bit; this is usually due to

5400-520: The time, however, most compilers used alternative representations for many of the operators, such as AND or 'AND' . This approach with BOOLEAN as a built-in (either primitive or otherwise predefined) data type was adopted by many later programming languages, such as Simula 67 (1967), ALGOL 68 (1970), Pascal (1970), Ada (1980), Java (1995), and C# (2000), among others. The first version of FORTRAN (1957) and its successor FORTRAN II (1958) have no logical values or operations; even

5475-532: The value true or false. The only operators that can accept operands of type bool are: &, |, ^, &=, |=, ^=, !, &&, || and ?:. A bool value can be implicitly converted to any integral type, with false becoming 0 and true becoming 1. The numeric literals 0 and 1 can be implicitly converted to the bool values false and true, respectively. Casting an expression to bool means testing for 0 or !=0 for arithmetic types, and null or !=null for pointers or references. Objective-C also has

5550-476: The ways computers transfer blocks of information. Most programming languages, even those with no explicit Boolean type, have support for Boolean algebraic operations such as conjunction ( AND , & , * ), disjunction ( OR , | , + ), equivalence ( EQV , = , == ), exclusive or /non-equivalence ( XOR , NEQV , ^ , != , ¬ ), and negation ( NOT , ~ , ! , ¬ ). In some languages, like Ruby , Smalltalk , and Alice

5625-525: Was then provided as a predefined enumerated type with values FALSE and TRUE . By definition, all comparisons, logical operations, and conditional statements applied to and/or yielded Boolean values. Otherwise, the Boolean type had all the facilities which were available for enumerated types in general, such as ordering and use as indices. In contrast, converting between Boolean s and integers (or any other types) still required explicit tests or function calls, as in ALGOL 60. This approach ( Boolean

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