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According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries.

the hands of thinkers such as George Boole, Gottlob Frege, Bertrand Russell, Alfred Tarski and Kurt Gödel, it’s clear that Kant was dead wrong. But he was also wrong in thinking that there had been no progress since Aristotle up to his time. <span>According to A History of Formal Logic (1961) by the distinguished J M Bocheński, the golden periods for logic were the ancient Greek period, the medieval scholastic period, and the mathematical period of the 19th and 20th centuries. (Throughout this piece, the focus is on the logical traditions that emerged against the background of ancient Greek logic. So Indian and Chinese logic are not included, but medieval Ara

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or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to <span>measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"

d-SEPARATION WITHOUT TEARS (At the request of many readers) Introduction d-separation is a criterion for deciding, from a given a causal graph, whether a set X of variables is independent of another set Y, given a third set Z. <span>The idea is to associate "dependence" with "connectedness" (i.e., the existence of a connecting path) and "independence" with "unconnected-ness" or "separation". The only twist on this simple idea is to define what we mean by "connecting path", given that we are dealing with a system of directed arrows in which some vertices (those residing in Z) correspond to measured variables, whose values are known precisely. To account for the orientations of the arrows we use the terms "d-separated" and "d-connected" (d connotes "directional"). We start by considering separation between two singleton variables, x and y; the extension to sets of variables is straightforward (i.e., two sets are separated if and only if each el

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tical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

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Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics.

tical space that allows for the definition of concepts such as continuity, connectedness, and convergence. [1] Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints. <span>Being so general, topological spaces are a central unifying notion and appear in virtually every branch of modern mathematics. The branch of mathematics that studies topological spaces in their own right is called point-set topology or general topology. Contents [hide] 1 History 2 Definition 2.1 De

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Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

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In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below.

Topological manifold - Wikipedia Topological manifold From Wikipedia, the free encyclopedia Jump to: navigation, search In topology, a branch of mathematics, a topological manifold is a topological space (which may also be a separated space) which locally resembles real n-dimensional space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topolog

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∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, <span>f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X. An extreme example: if a set X is giv

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intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology). <span>A function f : X → Y {\displaystyle f\colon X\rightarrow Y} between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image f − 1 ( V ) = { x ∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to

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A function between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image is an open subset of X.

intersections that generalize the properties of the open balls in metric spaces while still allowing to talk about the neighbourhoods of a given point. The elements of a topology are called open subsets of X (with respect to the topology). <span>A function f : X → Y {\displaystyle f\colon X\rightarrow Y} between two topological spaces X and Y is continuous if for every open set V ⊆ Y, the inverse image f − 1 ( V ) = { x ∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to

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In topological spaces, a function f is a function is defined on the sets X and Y, but the continuity of f depends on the topologies used on X and Y.

∈ X | f ( x ) ∈ V } {\displaystyle f^{-1}(V)=\{x\in X\;|\;f(x)\in V\}} is an open subset of X. That is, <span>f is a function between the sets X and Y (not on the elements of the topology T X ), but the continuity of f depends on the topologies used on X and Y. This is equivalent to the condition that the preimages of the closed sets (which are the complements of the open subsets) in Y are closed in X. An extreme example: if a set X is giv

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three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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A topological space is an ordered pair (X, τ), where X is a set and τ is a topology of X

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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ion of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. <span>The elements of τ are called open sets and the collection τ is called a topology on X. <span><body><html>

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ.

three-point set {1,2,3}. The bottom-left example is not a topology because the union of {2} and {3} [i.e. {2,3}] is missing; the bottom-right example is not a topology because the intersection of {1,2} and {2,3} [i.e. {2}], is missing. <span>A topological space is an ordered pair (X, τ), where X is a set and τ is a collection of subsets of X, satisfying the following axioms: [7] The empty set and X itself belong to τ. Any (finite or infinite) union of members of τ still belongs to τ. The intersection of any finite number of members of τ still belongs to τ. The elements of τ are called open sets and the collection τ is called a topology on X. Examples[edit source] Given X = {1, 2, 3, 4}, the collection τ = {{}, {1, 2, 3, 4}} of only the two subsets of X required by the axioms forms a topology of X, the trivial topology (

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ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

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A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets.

ected spaces 2 Examples 3 Path connectedness 4 Arc connectedness 5 Local connectedness 6 Set operations 7 Theorems 8 Graphs 9 Stronger forms of connectedness 10 See also 11 References 12 Further reading Formal definition[edit source] <span>A topological space X is said to be disconnected if it is the union of two disjoint nonempty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with it

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tyle \epsilon } less than half this distance, sequence terms cannot be within a distance ϵ {\displaystyle \epsilon } of both points. Topological spaces[edit source] Definition[edit source] <span>A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every n ≥ N {\displaystyle n\geq N} , x n ∈ U {\displaystyle x_{n}\in U} . This coincides with the definition given for metric spaces if (X,d) is a metric space and τ {\displaystyle \tau } is the topology generated b

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A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every , .

tyle \epsilon } less than half this distance, sequence terms cannot be within a distance ϵ {\displaystyle \epsilon } of both points. Topological spaces[edit source] Definition[edit source] <span>A point x of the topological space (X, τ) is the limit of the sequence (x n ) if, for every neighbourhood U of x, there is an N such that, for every n ≥ N {\displaystyle n\geq N} , x n ∈ U {\displaystyle x_{n}\in U} . This coincides with the definition given for metric spaces if (X,d) is a metric space and τ {\displaystyle \tau } is the topology generated b

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In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.

ogy (disambiguation). For a topology of a topos or category, see Lawvere–Tierney topology and Grothendieck topology. [imagelink] Möbius strips, which have only one surface and one edge, are a kind of object studied in topology. <span>In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important

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c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets. [3] [4] <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

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, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

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The Lebesgue outer measure of a set E emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets (unions of open intervals that include E).

, because E {\displaystyle E} is a subset of the union of the intervals, and so the intervals may include points which are not in E {\displaystyle E} . <span>The Lebesgue outer measure emerges as the greatest lower bound (infimum) of the lengths from among all possible such sets. Intuitively, it is the total length of those interval sets which fit E {\displaystyle E} most tightly and do not overlap. That characterize

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Any countable set of real numbers has Lebesgue measure 0.

c, d] is Lebesgue measurable, and its Lebesgue measure is (b − a)(d − c), the area of the corresponding rectangle. Moreover, every Borel set is Lebesgue measurable. However, there are Lebesgue measurable sets which are not Borel sets. [3] [4] <span>Any countable set of real numbers has Lebesgue measure 0. In particular, the Lebesgue measure of the set of rational numbers is 0, although the set is dense in R. The Cantor set is an example of an uncountable set that has Lebesgue measure zer

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ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

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In mathematics, the infimum of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists.

ed sets the infimum and the minimum are equal. [imagelink] A set A of real numbers (blue balls), a set of upper bounds of A (red diamond and balls), and the smallest such upper bound, that is, the supremum of A (red diamond). <span>In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set T is the greatest element in T that is less than or equal to all elements of S, if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used. The supremum (abbreviated sup; plural suprema) of a subset S of a partially ordered set T is th

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hieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). <span>More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective functio

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More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints.

hieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). <span>More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective functio

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ikipedia Crumpling From Wikipedia, the free encyclopedia Jump to: navigation, search "Crumpled" redirects here. For the deformation feature, see Crumple zone. <span>In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology. [1] Crumpled paper balls have been studied and found t

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In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges a

ikipedia Crumpling From Wikipedia, the free encyclopedia Jump to: navigation, search "Crumpled" redirects here. For the deformation feature, see Crumple zone. <span>In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a random network of ridges and facets with variable density. The geometry of crumpled structures is the subject of some interest the mathematical community within the discipline of topology. [1] Crumpled paper balls have been studied and found t

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integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, <span>the coefficient of δf in the first order term is called the functional derivative. For example, consider the functional J [ f ] = ∫ a

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the coefficient of arbitrarily small function δf in the first order term is called the functional derivative

integral of functions, their arguments, and their derivatives. In an integral L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, <span>the coefficient of δf in the first order term is called the functional derivative. For example, consider the functional J [ f ] = ∫ a

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>Cox's theorem - Wikipedia Cox's theorem From Wikipedia, the free encyclopedia (Redirected from Probability theory as extended logic) Jump to: navigation, search Cox's theorem, named after the physicist Richard Threlkeld Cox, is a derivation of the laws of probability theory from a certain set of postulates. This derivation justifies the so-called "logical" interpretation of probability, as the laws of probability derived by Cox's theorem are applicable to any proposition. Logical (a.k.a. objective Bayesian) probability is a type of Bayesian probability. Other forms of Bayesianism, such as the subjective interpretation, are given other justifications.

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ge language: English Brazilian Portuguese Chinese (Simplified) French German Japanese Romanian Russian Spanish Turkish Other Edit Report a Bug Escaping from HTML ¶ <span>Everything outside of a pair of opening and closing tags is ignored by the PHP parser which allows PHP files to have mixed content. This allows PHP to be embedded in HTML documents, for example to create templates. <span>

This is going to be ignored by PHP and displayed by the browser.<

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example to create templates.

This is going to be ignored by PHP and displayed by the browser.

This will also be ignored by PHP and displayed by the browser.

This works as expected, because when the PHP interpreter hits the ?> closing tags, it simply starts outputting whatever it finds (except for an immediately following newline - see instruction separation) until it hits another opening tag unless in the middle of a conditional statement in which case the interpreter will determine the outcome of tstatus | not read | reprioritisations | ||
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he condition is not met, even though they are outside of the PHP open/close tags; PHP skips them according to the condition since the PHP interpreter will jump over blocks contained within a condition that is not met. <span>For outputting large blocks of text, dropping out of PHP parsing mode is generally more efficient than sending all of the text through echo or print. In PHP 5, there are up to five different pairs of opening and closing tags available in PHP, depending on how PHP is configured. Two of these, and <

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blocks contained within a condition that is not met. For outputting large blocks of text, dropping out of PHP parsing mode is generally more efficient than sending all of the text through echo or print. <span>In PHP 5, there are up to five different pairs of opening and closing tags available in PHP, depending on how PHP is configured. Two of these, and <span> , are always available. There is also the short echo tag , w

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short echo tag , which is always available in PHP 5.4.0 and later.

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