formulasearchengine - User contributions [en]

Multiplication

2015-01-04T13:24:42Z

92.26.220.85: punctuation

In Mark 1:September 11 we learn that Jesus was baptized by John the Baptist in the Jordan River. During this event, the Holy Spirit descended on Jesus "like a dove" (Mark 1:10). Do you ever surprise why the Mark and the other gospel writers chose a dove to explain the coming of the Spirit upon Jesus? Let's check out the meaning of the dove at the baptism of Jesus. The most typical answer to this query is that the dove is an emblem of peace. Jesus is named the Prince of Peace in Isaiah 9:6. And on the night time of his betrayal, Jesus advised his disciples, "Peace I go away with you; my peace I offer you" (John 14:27). However I'm not so certain that's the best explanation of the which means of this. If you are you looking for more on [http://www.bible-unlimited.com/ French Mcarthur bible] visit the web site. Let's return to first century Israel and ask, "What would a Jewish individual think about when he saw a dove?" Peace? No. How about ache - the ache of a bloody animal sacrifice. When the Jews introduced an animal sacrifice to the temple to atone for his or her sins, the Previous Testament law supplied three options: a bull (for the rich), a lamb (for the middle class), and a dove (for the poor). There have been many poor folks in historic Israel, so it's likely that many or even most individuals would carry a dove to the priest because the sacrifice for sin. So on the day he was baptized, God the Father was saying to God the Son, "You are about to start your ministry, and you will spend the subsequent three years preaching the gospel, educating the Word, therapeutic individuals by the 1000's and performing miracles by no means earlier than seen on this planet. But the primary purpose you're right here, Jesus, is because you will die on the cross as a bloody sacrifice - similar to a dove - to pay the penalty for sins that responsible sinners deserve to pay." Is that this not the heart of biblical Christianity and one of the foundations of Bible doctrines? When John the Baptist noticed Jesus, he proclaimed for all to hear, "Look, the Lamb of God who takes away the sin of the world" (John 1:29). Jesus isn't solely the Lamb of God, he is additionally the Dove of God, and due to his loss of life, God causes our sins to fly away like a dove - if we but repent and belief in Jesus as Savior, Lord and Treasure. How far-off? "As far as the east is from the west, to this point has he eliminated our sins from us" (Psalm 103:12).

Linear code sequence and jump

2013-09-12T10:38:35Z

92.26.55.12: Consistency

The '''Föppl–von Kármán equations''', named after [[August Föppl]]<ref>Föppl, A., "Vorlesungen über technische Mechanik", ''B.G. Teubner'', Bd. 5., p. 132, Leipzig, Germany (1907)</ref> and [[Theodore von Kármán]],<ref>von Kármán, T., "Festigkeitsproblem im Maschinenbau," ''Encyk. D. Math. Wiss.'' '''IV''', 311–385 (1910)</ref> are a set of nonlinear [[partial differential equation]]s describing the large deflections of thin flat plates.<ref>E. Cerda and L. Mahadevan, 2003, "Geometry and Physics of Wrinkling" [http://prola.aps.org/abstract/PRL/v90/i7/e074302 Phys. Rev. Lett. 90, 074302 (2003)]</ref> With application ranging from the design of submarine hulls to the mechanical properties of cell wall,<ref>http://focus.aps.org/story/v27/st6</ref> the equations are notoriously difficult to solve, and take the following form:
<ref name="ld">"Theory of Elasticity". L. D. Landau, E. M. Lifshitz, (3rd ed. ISBN 0-7506-2633-X)</ref>

:<math>
\begin{align}
(1) \qquad & \frac{Eh^3}{12(1-\nu^2)}\Delta^2 w-h\frac{\partial}{\partial x_\beta}\left(\sigma_{\alpha\beta}\frac{\partial w}{\partial x_\alpha}\right)=P \\
(2) \qquad & \frac{\partial\sigma_{\alpha\beta}}{\partial x_\beta}=0
\end{align}
</math>

where {{math|''E''}} is the [[Young's modulus]] of the plate material (assumed homogeneous and isotropic), {{math|''υ''}} is the [[Poisson's ratio]], {{math|''h''}} is the thickness of the plate, {{math|''w''}} is the out–of–plane deflection of the plate, {{math|''P''}} is the external normal force per unit area of the plate, {{math|''σ''''αβ''}} is the [[Cauchy stress tensor]], and {{math|''α'', ''β''}} are [[Einstein notation|indices]] that take values of 1 or 2. The 2-dimensional [[biharmonic equation|biharmonic operator]] is defined as<ref>The 2-dimensional [[Laplacian]], {{math|Δ}}, is defined as
<math> \Delta w := \frac{\partial^2 w}{\partial x_\alpha \partial x_\alpha} = \frac{\partial^2w}{\partial x_1^2} + \frac{\partial^2w}{\partial x_2^2} </math></ref>
:<math>
\Delta^2 w := \frac{\partial^2}{\partial x_\alpha \partial x_\alpha}\left[\frac{\partial^2 w}{\partial x_\beta \partial x_\beta}\right]
= \frac{\partial^4 w}{\partial x_1^4} + \frac{\partial^4 w}{\partial x_2^4} + 2\frac{\partial^4 w}{\partial x_1^2 \partial x_2^2} \,.
</math>
Equation (1) above can be derived from [[kinematic]] assumptions and the [[constitutive relation]]s for the plate. Equations (2) are the two equations for the conservation of linear momentum in two dimensions where it is assumed that the out–of–plane stresses ({{math|''σ''33,''σ''13,''σ''23}}) are zero.

== Validity of the Föppl–von Kármán equations ==
While the Föppl–von Kármán equations are of interest from a purely mathematical point of view, the physical validity of these equations is questionable.<ref>[http://imechanica.org/node/6618 von Karman plate equations http://imechanica.org/node/6618 Accessed Tue July 30 2013 14:20.]</ref> Ciarlet<ref name=Ciarlet>{{Citation|last=Ciarlet|first=P. G.|year=1990|title=Plates and Junctions in Elastic Multi-Structures|publisher=Springer-Verlag.}}</ref> states: ''The two-dimensional von Karman equations for plates, originally proposed by von Karman [1910], play a mythical role in applied mathematics. While they have been abundantly, and satisfactorily, studied from the mathematical standpoint, as regards notably various questions of existence, regularity, and bifurcation, of their solutions, their physical soundness has been often seriously questioned.'' Reasons include the facts that
# the theory depends on an approximate geometry which is not clearly defined
# a given variation of stress over a cross-section is assumed arbitrarily
# a linear constitutive relation is used that does not correspond to a known relation between well defined measures of stress and strain
# some components of strain are arbitrarily ignored
# there is a confusion between reference and deformed configurations which makes the theory inapplicable to the large deformations for which it was apparently devised.
Conditions under which these equations are actually applicable and will give reasonable results when solved are discussed in Ciarlet.<ref name=Ciarlet/><ref>{{Citation|last= Ciarlet|first= Philippe G. | title=A justification of the von Kármán equations|journal= Archive for Rational Mechanics and Analysis |volume=73|year=1980|pages= 349–389.|issue=4|bibcode= 1980ArRMA..73..349C|doi= 10.1007/BF00247674}}</ref>

== Equations in terms of Airy stress function ==
The three Föppl–von Kármán equations can be reduced to two by introducing the [[Stress functions#Airy stress function|Airy stress function]] <math>\varphi</math> where
:<math>
\sigma_{11} = \frac{\partial^2 \varphi}{\partial x_2^2} ~,~~
\sigma_{22} = \frac{\partial^2 \varphi}{\partial x_1^2} ~,~~
\sigma_{12} = - \frac{\partial^2 \varphi}{\partial x_1 \partial x_2} \,.
</math>
Then the above equations become<ref name="ld"/>
:<math>
\frac{Eh^3}{12(1-\nu^2)}\Delta^2 w-h\left(\frac{\partial^2\varphi}{\partial x_2^2}\frac{\partial^2 w}{\partial x_1^2}+\frac{\partial^2\varphi}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-2\frac{\partial^2\varphi}{\partial x_1 \, \partial x_2}\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)=P
</math>
:<math>
\Delta^2\varphi+E\left\{\frac{\partial^2 w}{\partial x_1^2}\frac{\partial^2 w}{\partial x_2^2}-\left(\frac{\partial^2 w}{\partial x_1 \, \partial x_2}\right)^2\right\}=0 \,.
</math>

== Pure bending==
For the [[pure bending]] of thin plates the equation of equilibrium is <math>D\Delta^2\ w=P</math>, where
:<math>
D :=\frac{Eh^3}{12(1-\nu^2)}
</math>
is called [[flexural rigidity|flexural]] or ''cylindrical rigidity'' of the plate.<ref name="ld"/>

== Kinematic assumptions (Kirchhoff hypothesis) ==
In the derivation of the Föppl–von Kármán equations the main kinematic assumption (also known as the '''Kirchhoff hypothesis''') is that [[surface normal]]s to the plane of the plate remain perpendicular to the plate after deformation. It is also assumed that the in-plane (membrane) displacements and the change in thickness of the plate are negligible. These assumptions imply that the displacement field {{math|'''u'''}} in the plate can be expressed as
:<math>
u_1(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_1} ~,~~
u_2(x_1,x_2,x_3) = -x_3\,\frac{\partial w}{\partial x_2} ~,~~
u_3(x_1, x_2, x_3) = w(x_1,x_2)
</math>
This form of the displacement field implicitly assumes that the amount of rotation of the plate is small.

== Strain-displacement relations (von Kármán strains) ==
The components of the three-dimensional Lagrangian [[Finite strain theory|Green strain tensor]] are defined as
:<math>
E_{ij} := \frac{1}{2}\left[\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}
+ \frac{\partial u_k}{\partial x_i}\,\frac{\partial u_k}{\partial x_j}\right] \,.
</math>
Substitution of the expressions for the displacement field into the above gives
:<math>
\begin{align}
E_{11} & = \frac{\partial u_1}{\partial x_1}
+ \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_1}\right)^2
+ \left(\frac{\partial u_2}{\partial x_1}\right)^2
+ \left(\frac{\partial u_3}{\partial x_1}\right)^2\right]\\
&= -x_3\,\frac{\partial^2 w}{\partial x_1^2}
+ \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)^2
+ x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2
+ \left(\frac{\partial w}{\partial x_1}\right)^2\right]\\
E_{22} & = \frac{\partial u_2}{\partial x_2}
+ \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_2}\right)^2
+ \left(\frac{\partial u_2}{\partial x_2}\right)^2
+ \left(\frac{\partial u_3}{\partial x_2}\right)^2\right]\\
&= -x_3\,\frac{\partial^2 w}{\partial x_2^2}
+ \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)^2
+ x_3^2\left(\frac{\partial^2 w}{\partial x_2^2}\right)^2
+ \left(\frac{\partial w}{\partial x_2}\right)^2\right]\\
E_{33} & = \frac{\partial u_3}{\partial x_3}
+ \frac{1}{2}\left[\left(\frac{\partial u_1}{\partial x_3}\right)^2
+ \left(\frac{\partial u_2}{\partial x_3}\right)^2
+ \left(\frac{\partial u_3}{\partial x_3}\right)^2\right]\\
&= \frac{1}{2}\left[\left(\frac{\partial w}{\partial x_1}\right)^2
+ \left(\frac{\partial w}{\partial x_2}\right)^2
\right]\\
E_{12} & = \frac{1}{2}\left[\frac{\partial u_1}{\partial x_2} + \frac{\partial u_2}{\partial x_1}
+ \frac{\partial u_1}{\partial x_1}\,\frac{\partial u_1}{\partial x_2}
+ \frac{\partial u_2}{\partial x_1}\,\frac{\partial u_2}{\partial x_2}
+ \frac{\partial u_3}{\partial x_1}\,\frac{\partial u_3}{\partial x_2}\right]\\
& = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
+ \frac{1}{2}\left[x_3^2\left(\frac{\partial^2 w}{\partial x_1^2}\right)\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right)
+ x_3^2\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_2^2}\right)
+ \frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right]\\
E_{23} & = \frac{1}{2}\left[\frac{\partial u_2}{\partial x_3} + \frac{\partial u_3}{\partial x_2}
+ \frac{\partial u_1}{\partial x_2}\,\frac{\partial u_1}{\partial x_3}
+ \frac{\partial u_2}{\partial x_2}\,\frac{\partial u_2}{\partial x_3}
+ \frac{\partial u_3}{\partial x_2}\,\frac{\partial u_3}{\partial x_3}\right]\\
& = \frac{1}{2}\left[x_3\left(\frac{\partial^2 w}{\partial x_1\partial x_2}\right)\left(\frac{\partial w}{\partial x_1}\right)
+ x_3\left(\frac{\partial^2 w}{\partial x_2^2}\right)\left(\frac{\partial w}{\partial x_2}\right)
\right]\\
E_{31} & = \frac{1}{2}\left[\frac{\partial u_3}{\partial x_1} + \frac{\partial u_1}{\partial x_3}
+ \frac{\partial u_1}{\partial x_3}\,\frac{\partial u_1}{\partial x_1}
+ \frac{\partial u_2}{\partial x_3}\,\frac{\partial u_2}{\partial x_1}
+ \frac{\partial u_3}{\partial x_3}\,\frac{\partial u_3}{\partial x_1}\right] \\
& = \frac{1}{2}\left[x_3\left(\frac{\partial w}{\partial x_1}\right)\left(\frac{\partial^2 w}{\partial x_1^2}\right)
+ x_3\left(\frac{\partial w}{\partial x_2}\right)\left(\frac{\partial^2 w}{\partial x_1 \partial x_2}\right)
\right]
\end{align}
</math>
For small strains but '''moderate rotations''', the higher order terms that cannot be neglected are
:<math>
\left(\frac{\partial w}{\partial x_1}\right)^2 ~,~~ \left(\frac{\partial w}{\partial x_2}\right)^2 ~,~~
\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2} \,.
</math>
Neglecting all other higher order terms, and enforcing the requirement that the plate does not change its thickness, the strain tensor components reduce to the '''von Kármán strains'''
:<math>
\begin{align}
E_{11} & = -x_3\,\frac{\partial^2 w}{\partial x_1^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \\
E_{22} & = -x_3\,\frac{\partial^2 w}{\partial x_2^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \\
E_{12} & = -x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
+ \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\\
E_{33} & = 0 ~,~~ E_{23} = 0 ~,~~ E_{31} = 0 \,.
\end{align}
</math>

== Stress-strain relations ==
If we assume that the [[Cauchy stress tensor]] components are linearly related to the von Kármán strains by [[Hooke's law]], the plate is isotropic and homogeneous, and that the plate in under a [[plane stress]] condition,<ref>Typically, an assumption of '''zero out-of-plane stress''' is made at this point.</ref> we have {{math|''σ''33}} = {{math|''σ''13}} = {{math|''σ''23}} = 0 and
:<math>
\begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{12} \end{bmatrix}
= \cfrac{E}{(1-\nu^2)}
\begin{bmatrix} 1 & \nu & 0 \\
\nu & 1 & 0 \\
0 & 0 & 1-\nu \end{bmatrix}
\begin{bmatrix} E_{11} \\ E_{22} \\ E_{12} \end{bmatrix}
</math>
Expanding the terms, the three non-zero stresses are
:<math>
\begin{align}
\sigma_{11} &= \cfrac{E}{(1-\nu^2)}\left[\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) +
\nu\left(-x_3\,\frac{\partial^2 w}{\partial x_2^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\
\sigma_{22} &= \cfrac{E}{(1-\nu^2)}\left[\nu\left(-x_3\,\frac{\partial^2 w}{\partial x_1^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_1}\right)^2 \right) +
\left(-x_3\,\frac{\partial^2 w}{\partial x_2^2}
+ \frac{1}{2}\left(\frac{\partial w}{\partial x_2}\right)^2 \right) \right] \\
\sigma_{12} &= \cfrac{E}{(1+\nu)}\left[-x_3\frac{\partial^2 w}{\partial x_1 \partial x_2}
+ \frac{1}{2}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}\right] \,.
\end{align}
</math>

== Stress resultants ==
The [[stress resultants]] in the plate are defined as
:<math>
N_{\alpha\beta} := \int_{-h/2}^{h/2} \sigma_{\alpha\beta}\, d x_3 ~,~~
M_{\alpha\beta} := \int_{-h/2}^{h/2} x_3\,\sigma_{\alpha\beta}\, d x_3 \,.
</math>
Therefore,
:<math>
\begin{align}
N_{11} &= \cfrac{Eh}{2(1-\nu^2)}\left[\left(\frac{\partial w}{\partial x_1}\right)^2
+ \nu\left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\
N_{22} &= \cfrac{Eh}{2(1-\nu^2)}\left[\nu\left(\frac{\partial w}{\partial x_1}\right)^2
+ \left(\frac{\partial w}{\partial x_2}\right)^2 \right] \\
N_{12} &= \cfrac{Eh}{2(1+\nu)}\,\frac{\partial w}{\partial x_1}\,\frac{\partial w}{\partial x_2}
\end{align}
</math>
and
:<math>
\begin{align}
M_{11} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\frac{\partial^2 w}{\partial x_1^2} +\nu \,\frac{\partial^2 w}{\partial x_2^2} \right] \\
M_{22} &= -\cfrac{Eh^3}{12(1-\nu^2)}\left[\nu \,\frac{\partial^2 w}{\partial x_1^2} +\frac{\partial^2 w}{\partial x_2^2} \right] \\
M_{12} &= -\cfrac{Eh^3}{12(1+\nu)}\,\frac{\partial^2 w}{\partial x_1 \partial x_2} \,.
\end{align}
</math>
Solutions are easier to find when the governing equations are expressed in terms of stress resultants rather than the in-plane stresses.

== Föppl–von_Kármán equations in terms of stress resultants ==
The Föppl–von_Kármán equations are typically derived with an energy approach by considering [[variational calculus|variation]]s of internal energy and the work done by external forces. A similar approach can be used to write these equations in terms of stress resultants. The resulting governing equations are
:<math>
\begin{align}
&\frac{\partial^2 M_{11}}{\partial x_1^2} + \frac{\partial^2 M_{22}}{\partial x_2^2} + 2\frac{\partial^2 M_{12}}{\partial x_1\partial x_2} +
\frac{\partial}{\partial x_1}\left(N_{11}\,\frac{\partial w}{\partial x_1} + N_{12}\,\frac{\partial w}{\partial x_2}\right) +
\frac{\partial}{\partial x_2}\left(N_{12}\,\frac{\partial w}{\partial x_1} + N_{22}\,\frac{\partial w}{\partial x_2}\right) = P \\
& \frac{\partial N_{\alpha\beta}}{\partial x_\beta} = 0 \,.
\end{align}
</math>

==References==
<references/>

== See also ==
* [[Plate theory]]

{{DEFAULTSORT:Foppl-Von Karman Equations}}
[[Category:Partial differential equations]]
[[Category:Continuum mechanics]]

List of recurring Futurama characters

2012-09-02T16:24:48Z

92.26.76.27: /* Officer Smitty */

I’m Keith from Sankt Stefan Ob Stainz studying Africana Studies. I did my schooling, secured 91% and hope to find someone with same interests in Stone collecting. Also visit my homepage [http://linxfix.com/WordpressBackupPlugin477187 wordpress backup]

Utm theorem

2012-04-02T21:13:48Z

92.26.62.114:

{{Transformation rules}}

In [[propositional calculus|propositional logic]], '''transposition'''<ref>{{cite book |title=A Concise Introduction to Logic 4th edition |last=Hurley |first=Patrick |authorlink= |coauthors= |year=1991 |publisher=Wadsworth Publishing |location= |isbn= |page= |pages=364–5 |url= |accessdate=}}</ref>{{verify source|date=February 2012}}<ref>{{cite book |ref=harv |last=Copi |first=Irving M. |last2=Cohen |first2=Carl |title=Introduction to Logic |publisher=Prentice Hall |year=2005 |page=371 |isbn=}}</ref><ref>Moore and Parker</ref> is a [[validity|valid]] [[rule of replacement]] that permits one to switch the [[antecedent (logic)|antecedent]] with the [[consequent]] of a [[material conditional|conditional statement]] in a [[formal proof|logical proof]] if they are also both [[logical negation|negated]]. It is the [[inference]] from the truth of "''A'' implies ''B''" the truth of "Not-''B'' implies not-''A''", and conversely.<ref>Brody, Bobuch A. "Glossary of Logical Terms". ''Encyclopedia of Philosophy''. Vol. 5–6, p. 76. Macmillan, 1973.</ref><ref>Copi, Irving M. ''Symbolic Logic''. 5th ed. Macmillan, 1979. See the Rules of Replacement, pp. 39-40.</ref> It is very closely related to the [[rule of inference]] [[modus tollens]]. It is the rule that:

:(''P'' {{imp}} ''Q'') <math>\Leftrightarrow</math> ({{not}} ''Q'' {{imp}} {{not}} ''P'')

Where "<math>\Leftrightarrow</math>" is a [[metalogic]]al [[Symbol (formal)|symbol]] representing "can be replaced in a proof with."

== Formal notation ==
The ''transposition'' rule may be expressed as a [[sequent]]:
:<math>(P \to Q) \vdash (\neg Q \to \neg P)</math>

where <math>\vdash</math> is a metalogical symbol meaning that <math>(\neg Q \to \neg P)</math> is a [[logical consequence|syntactic consequence]] of <math>(P \to Q)</math> in some logical system;

or as a rule of inference:
:<math>\frac{P \to Q}{\therefore \neg Q \to \neg P}</math>
where the rule is that wherever an instance of "<math>P \to Q</math>" appears on a line of a proof, it can be replaced with "<math>\neg Q \to \neg P</math>";

or as the statement of a truth-functional [[Tautology (logic)|tautology]] or [[theorem]] of propositional logic. The principle was stated as a theorem of propositional logic by [[Bertrand Russell|Russell]] and [[Alfred North Whitehead|Whitehead]] in ''[[Principia Mathematica]]'' as:

:<math>(P \to Q) \to (\neg Q \to \neg P)</math>

where <math>P</math> and <math>Q</math> are propositions expressed in some [[formal system]].

==Traditional logic==
=== Form of transposition===
In the inferred proposition, the consequent is the contradictory of the antecedent in the original proposition, and the antecedent of the inferred proposition is the contradictory of the consequent of the original proposition. The symbol for material implication signifies the proposition as a hypothetical, or the "if-then" form, e.g. "if P then Q".

The biconditional statement of the rule of transposition (↔) refers to the relation between hypothetical (→) ''propositions'', with each proposition including an antecent and consequential term. As a matter of logical inference, to transpose or convert the terms of one proposition requires the conversion of the terms of the propositions on both sides of the biconditional relationship. Meaning, to transpose or convert (P → Q) to (Q → P) requires that the other proposition, (~Q → ~P), be transposed or converted to (~P → ~Q). Otherwise, to convert the terms of one proposition and not the other renders the rule invalid, violating the [[sufficient condition]] and [[necessary condition]] of the terms of the propositions, where the violation is that the changed proposition commits the fallacy of [[denying the antecedent]] or [[affirming the consequent]] by means of illicit [[Conversion (logic)|conversion]]

The truth of the rule of transposition is dependent upon the relations of sufficient condition and necessary condition in logic.

===Sufficient condition===
In the proposition "If P then Q", the occurrence of 'P' is sufficient reason for the occurrence of 'Q'. 'P', as an individual or a class, materially implicates 'Q', but the relation of 'Q' to 'P' is such that the converse proposition "If Q then P" does not necessarily have sufficient condition. The rule of inference for sufficient condition is ''modus ponens'', which is an argument for conditional implication:

Premise (1): If P, then Q

Premise (2): P

Conclusion: Therefore, Q

===Necessary condition===
Since the converse of premise (1) is not valid, all that can be stated of the relationship of 'P' and 'Q' is that in the absence of 'Q', 'P' does not occur, meaning that 'Q' is the necessary condition for 'P'. The rule of inference for necessary condition is ''modus tollens'':

Premise (1): If P, then Q

Premise (2): not Q

Conclusion: Therefore, not P

===Grammatically speaking===
A grammatical example traditionally used by logicians contrasting sufficient and necessary conditions is the statement "If there is fire, then oxygen is present". An oxygenated environment is necessary for fire or combustion, but simply because there is an oxygenated environment does not necessarily mean that fire or combustion is occurring. While one can infer that fire stipulates the presence of oxygen, from the presence of oxygen the converse "If there is oxygen present, then fire is present" cannot be inferred. All that can be inferred from the original proposition is that "If oxygen is not present, then there cannot be fire".

===Relationship of propositions===
The symbol for the biconditional ("↔") signifies the relationship between the propositions is both necessary and sufficient, and is verbalized as "[[if and only if]]", or, according to the example "If P then Q 'if and only if' if not Q then not P".

Necessary and sufficient conditions can be explained by analogy in terms of the concepts and the rules of immediate inference of traditional logic. In the categorical proposition "All S is P", the subject term 'S' is said to be distributed, that is, all members of its class are exhausted in its expression. Conversely, the predicate term 'P' cannot be said to be distributed, or exhausted in its expression because it is indeterminate whether every instance of a member of 'P' as a class is also a member of 'S' as a class. All that can be validly inferred is that "Some P are S". Thus, the type 'A' proposition "All P is S" cannot be inferred by conversion from the original 'A' type proposition "All S is P". All that can be inferred is the type "A" proposition "All non-P is non-S" (Note that (P → Q) and (~Q → ~P) are both 'A' type propositions). Grammatically, one cannot infer "all mortals are men" from "All men are mortal". An 'A' type proposition can only be immediately inferred by conversion when both the subject and predicate are distributed, as in the inference "All bachelors are unmarried men" from "All unmarried men are bachelors".

===Transposition and the method of contraposition===
In [[traditional logic]] the reasoning process of transposition as a rule of inference is applied to [[categorical propositions]] through [[contraposition]] and [[obversion]],<ref>Stebbing, 1961, p. 65-66. For reference to the initial step of contraposition as obversion and conversion, see Copi, 1953, p. 141.</ref> a series of immediate inferences where the rule of obversion is first applied to the original categorical proposition "All S is P"; yielding the obverse "No S is non-P". In the obversion of the original proposition to an 'E' type proposition, both terms become distributed. The obverse is then converted, resulting in "No non-P is S", maintaining distribution of both terms. The No non-P is S" is again obverted, resulting in the [contrapositive] "All non-P is non-S". Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory, and the predicate term of the resulting 'A' type proposition is again undistributed. This results in two contrapositives, one where the predicate term is distributed, and another where the predicate term is undistributed.<ref>See Stebbing, 1961, pp. 65-66. Also, for reference to the immediate inferences of obversion, conversion, and obversion again, see Copi, 1953, p. 141.</ref>

===Differences between transposition and contraposition===
Note that the method of transposition and contraposition should not be confused. Contraposition is a type of [[immediate inference]] in which from a given categorical proposition another categorical proposition is inferred which has as its subject the contradictory of the original predicate. Since nothing is said in the definition of contraposition with regard to the predicate of the inferred proposition, it is permissible that it could be the original subject or its contradictory. This is in contradistinction to the form of the propositions of transposition, which may be material implication, or a hypothetical statement. The difference is that in its application to categorical propositions the result of contraposition is two contrapositives, each being the obvert of the other,<ref>See Stebbing, 1961, p. 66.</ref> i.e. "No non-P is S" and "All non-P is non-S". The distinction between the two contrapositives is absorbed and eliminated in the principle of transposition, which presupposes the "mediate inferences"<ref>For an explanation of the absorption of obversion and conversion as "mediate inferences see: Copi, Irving. ''Symbolic Logic''. pp. 171-174, MacMillan, 1979, fifth edition.</ref> of contraposition and is also referred to as the "law of contraposition".<ref>Prior, A.N. "Logic, Traditional". ''Encyclopedia of Philosophy'', Vol.5, Macmillan, 1973.</ref>

==Transposition in mathematical logic==
See [[Transposition (mathematics)]], [[Set theory]]

==Proof==
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:45%"
|+ ''' '''
|- style="background:paleturquoise"
! style="width:15%" | ''Proposition''
! style="width:15%" | ''Derivation''
|-
| <math>P\rightarrow Q</math> || Given
|-
| <math>\neg P\or Q</math> || [[Material implication (rule of inference)|Material implication]]
|-
| <math>Q\or\neg P</math> || [[Commutative property|Commutavity]]
|-
| <math>\neg Q\rightarrow\neg P</math> || Material implication
|}

==See also==
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{{col-break}}
*[[Contraposition (traditional logic)]]
{{col-break}}
*[[Syllogism]]
*[[Term logic]]
{{col-end}}

==References==
{{reflist}}

==Further reading==
*Brody, Bobuch A. "Glossary of Logical Terms". Encyclopedia of Philosophy. Vol. 5-6, p. 61. Macmillan, 1973.
*Copi, Irving. ''Introduction to Logic''. MacMillan, 1953.
*Copi, Irving. ''Symbolic Logic''. MacMillan, 1979, fifth edition.
*Prior, A.N. "Logic, Traditional". ''Encyclopedia of Philosophy'', Vol.5, Macmillan, 1973.
*[[Susan Stebbing|Stebbing, Susan]]. ''A Modern Introduction to Logic''. Harper, 1961, Seventh edition

==External links==
*[http://www.fallacyfiles.org/imptrans.html Improper Transposition] (Fallacy Files)

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[[Category:Rules of inference]]
[[Category:Theorems in propositional logic]]