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any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

es in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, <span>any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. Contents [hide] 1 Mathematical definition 1.1 Alternative definition 2 Representation 2.1 Augmented matrix 2.1.1 Example augmented matrix 3 Properties 3.1 Properti

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Verbs ending in -cer add a cedilla to the c before the letters a or o in order to retain the soft c sound. avancer to advance nous avanc¸ ons

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Verbs ending in -ger add an e after the g before the letters a and o in order to maintain the soft g sound. changer to change nous changeons

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Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except the nous and vous forms. compléter : je complète; tu complètes; il, elle, on complète; nous complétons; vous compléte

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Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except the nous and vous forms. compléter : je complète; tu complètes; il, elle, on complète; nous complétons; vous complétez; ils, elles complètent

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In other verbs with -e- in the infinitive, the final consonant is doubled in all but the nous and vous forms. jeter: je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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In other verbs with -e- in the infinitive, the final consonant is doubled in all but the nous and vous forms. jeter: je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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Verbs whose infinitive ends in -oyer, -uyer and -ayer change -y- to -i- in all but the nous and vous forms. nettoyer: je nettoie; tu nettoies; il, elle, on nettoie; nous nettoyons; vous nettoyez; ils, elles nettoient

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Verbs whose infinitive ends in -ir add -is, -is, -it, -issons, -issez, -issent to the stem. finir : je finis; tu finis; il, elle, on finit; nous finissons; vous finissez; ils, elles finissent

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Verbs whose infinitive ends in -re add -s, -s, -, -ons, -ez, -ent to the stem. répondre : je réponds; tu réponds; il, elle, on répond; nous répondons; vous répondez; ils, elles répondent

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Some verbs like ouvrir , offrir , and cueillir , although the infinitive ends in -ir, are conjugated like regular -er verbs.

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Some verbs like ouvrir , offrir , and cueillir , although the infinitive ends in -ir, are conjugated like regular -er verbs.

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, et

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, et

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

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the logical product or the conjunction, denotes the proposition ‘both A and B are true

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the logical sum or disjunction, stands for ‘at least one of the propositions, A, B is true’

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that A and B have the same truth value, then they are logically equivalent propositions, in the sense that any evidence concerning the truth of one pertains equally well to the truth of the other, and they have the same implications for any further reasoning

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The relation between a proposition and its denial is reciprocal

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The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic.

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The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic.

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The theory of plausible reasoning is based on weak syllogisms

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Plausible reasoning is an extension of deductive logic because it also applies to weak syllogisms.

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Plausible reasoning is an extension of deductive logic because it also applies to weak syllogisms.

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a false proposition implies all propositions

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merely knowing the truth values of propositions A and B does not provide any information on deducibility

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merely knowing the truth values of propositions A and B does not provide any information on deducibility

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logic functions with only one point being true can be constructed simply by conjunctions of input propositions,

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Any logic function that is true on at least one point can be constructed as logical sum of basic conjunctions

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The operation ‘NAND’ is defined as the negation of ‘AND’

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the operation NOR defined by the negation of OR

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the two operations (AND, NOT) already constitute an adequate set for deductive logic

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A set of operations suffice to generate all possible logic functions form an adequate set

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the two operations (AND, NOT) already constitute an adequate set for deductive logic

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A set of operations suffice to generate all possible logic functions form an adequate set

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besides a common ground, a logic gate has two input terminals and one output

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We expect plausible reasoning to have no contradictions and be able to determine any unique extension of logic

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We expect plausible reasoning to have no contradictions and be able to determine any unique extension of logic

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A ⇒ B is false only if A is true and B is false

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The question of logical deducibility of one proposition from a set of others arises in a crucial way in the Gödel theorem

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The duality property shows the logic disjunction A+B is the same as denying that they are both false A¯B¯⎯⎯⎯⎯⎯⎯⎯

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This great difference in the meaning of the word implies in ordinary language and in formal logic is a tricky point that can lead to serious error if it is not properly understood

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The duality property shows the logic disjunction A+B is the same as denying that they are both false A¯B¯⎯⎯⎯⎯⎯⎯⎯

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