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2014-01-21T22:37:24Z

173.73.151.72:

{{Portal|Logic}}
This is a list of [[Rule of inference|rules of inference]], logical laws that relate to mathematical formulae.

==Introduction==

'''Rules of inference''' are syntactical '''transform''' rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.

''Discharge rules'' permit inference from a subderivation based on a temporary assumption. Below, the notation

: <math>\varphi \vdash \psi\,\!</math>

indicates such a subderivation from the temporary assumption <math>\varphi\,\!</math> to <math>\psi\,\!</math>.

==Rules for classical sentential calculus==
Sentential calculus is also known as [[propositional calculus]].

===Rules for negations===
;[[Reductio ad absurdum]] (or ''Negation Introduction''):
: <math>\varphi \vdash \psi\,\!</math>
: <math>\underline{\varphi \vdash \lnot \psi}\,\!</math>
: <math>\lnot \varphi\,\!</math>

;Reductio ad absurdum (related to the law of [[excluded middle]]):
: <math>\lnot \varphi \vdash \psi\,\!</math>
: <math>\underline{\lnot \varphi \vdash \lnot \psi}\,\!</math>
: <math>\varphi\,\!</math>

;[[Noncontradiction]] (or ''Negation Elimination''):
: <math>\varphi\,\!</math>
: <math>\underline{\lnot \varphi}\,\!</math>
: <math>\psi\,\!</math>

;[[Double negative elimination|Double negation elimination]]:
: <math>\underline{\lnot \lnot \varphi}\,\!</math>
: <math> \varphi\,\!</math>

;[[Double negative introduction|Double negation introduction]]:
: <math>\underline{\varphi \quad \quad}\,\!</math>
: <math> \lnot \lnot \varphi\,\!</math>

===Rules for conditionals===
;[[Deduction theorem]] (or ''[[Conditional proof|Conditional Introduction]]''):
: <math>\underline{\varphi \vdash \psi}\,\!</math>
: <math>\varphi \rightarrow \psi\,\!</math>

;[[Modus ponens]] (or ''Conditional Elimination''):
: <math>\varphi \rightarrow \psi\,\!</math>
: <math>\underline{\varphi \quad \quad \quad}\,\!</math>
: <math>\psi\,\!</math>

;[[Modus tollens]]:
: <math>\varphi \rightarrow \psi\,\!</math>
: <math>\underline{\lnot \psi \quad \quad \quad}\,\!</math>
: <math>\lnot \varphi\,\!</math>

===Rules for conjunctions===
;[[Conjunction introduction|Adjunction]] (or ''Conjunction Introduction''):

: <math>\varphi\,\!</math>
: <math>\underline{\psi \quad \quad \ \ }\,\!</math>
: <math>\varphi \land \psi\,\!</math>

;[[Simplification (logic)|Simplification]] (or ''Conjunction Elimination''):

: <math>\underline{\varphi \land \psi}\,\!</math>
: <math>\varphi\,\!</math>

: <math>\underline{\varphi \land \psi}\,\!</math>
: <math>\psi\,\!</math>

===Rules for disjunctions===
;[[Addition (logic)|Addition]] (or ''Disjunction Introduction''):
: <math>\underline{\varphi \quad \quad \ \ }\,\!</math>
: <math>\varphi \lor \psi\,\!</math>

: <math>\underline{\psi \quad \quad \ \ }\,\!</math>
: <math>\varphi \lor \psi\,\!</math>

;[[Case analysis]]
: <math>\varphi \lor \psi\,\!</math>
: <math>\varphi \rightarrow \chi\,\!</math>
: <math>\underline{\psi \rightarrow \chi}\,\!</math>
: <math>\chi\,\!</math>

;[[Disjunctive syllogism]]:
: <math>\varphi \lor \psi\,\!</math>
: <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
: <math>\psi\,\!</math>

: <math>\varphi \lor \psi\,\!</math>
: <math>\underline{\lnot \psi \quad \quad}\,\!</math>
: <math>\varphi\,\!</math>

===Rules for biconditionals===

;[[Biconditional introduction]]:
: <math>\varphi \rightarrow \psi\,\!</math>
: <math>\underline{\psi \rightarrow \varphi}\,\!</math>
: <math>\varphi \leftrightarrow \psi\,\!</math>

;Biconditional Elimination:
: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\varphi \quad \quad}\,\!</math>
: <math>\psi\,\!</math>

: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\psi \quad \quad}\,\!</math>
: <math>\varphi\,\!</math>

: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\lnot \varphi \quad \quad}\,\!</math>
: <math>\lnot \psi\,\!</math>

: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\lnot \psi \quad \quad}\,\!</math>
: <math>\lnot \varphi\,\!</math>

: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\psi \lor \varphi}\,\!</math>
: <math>\psi \land \varphi \,\!</math>

: <math>\varphi \leftrightarrow \psi\,\!</math>
: <math>\underline{\lnot \psi \lor \lnot \varphi}\,\!</math>
: <math>\lnot \psi \land \lnot \varphi \,\!</math>

==Rules of classical [[First-order logic|predicate calculus]]==

In the following rules, <math>\varphi(\beta / \alpha)\,\!</math> is exactly like <math>\varphi\,\!</math> except for having the term <math>\beta\,\!</math> everywhere <math>\varphi\,\!</math> has the free variable <math>\alpha\,\!</math>.

;[[Universal generalization|Universal Introduction]] (or ''Universal Generalization''):
: <math>\underline{\varphi{(\beta / \alpha)}}\,\!</math>
: <math>\forall \alpha\, \varphi\,\!</math>

Restriction 1: <math>\beta</math> does not occur in <math>\varphi</math>.
 
Restriction 2: <math>\beta</math> is not mentioned in any hypothesis or undischarged assumptions.

;[[Universal instantiation|Universal Elimination]] (or ''Universal Instantiation''):
: <math> \forall \alpha\, \varphi\!</math>
: <math>\overline{\varphi{(\beta / \alpha)}}\!</math>

Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.

;Existential Introduction (or ''Existential Generalization''):
: <math>\underline{\varphi(\beta / \alpha)}\,\!</math>
: <math>\exists \alpha\, \varphi\,\!</math>

Restriction: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.

;Existential Elimination (or ''Existential Instantiation''):
: <math>\exists \alpha\, \varphi\,\!</math>
: <math>\underline{\varphi(\beta / \alpha) \vdash \psi}\,\!</math>
: <math>\psi\,\!</math>

Restriction 1: No free occurrence of <math>\alpha\,\!</math> in <math>\varphi\,\!</math> falls within the scope of a quantifier quantifying a variable occurring in <math>\beta\,\!</math>.
 
Restriction 2: There is no occurrence, free or bound, of <math>\beta\,\!</math> in <math>\psi\,\!</math>.

==Table: Rules of Inference - a short summary==
The rules above can be summed up in the following table.<ref>Kenneth H. Rosen: ''Discrete Mathematics and its Applications'',Fifth Edition, p. 58.</ref> The "[[Tautology (logic)|Tautology]]" column shows how to interpret the notation of a given rule.

{| class="wikitable"
|-
! Rule of inference
! Tautology
! Name
|-
|<math>\begin{align}
p \\
\therefore \overline{p \vee q} \\
\end{align}</math>
| <math>p \rightarrow (p \vee q)</math>
| Addition
|-
|<math>\begin{align}
p \wedge q \\
\therefore \overline{p \quad \quad \quad} \\
\end{align}</math>
| <math>(p \wedge q) \rightarrow p</math>
| Simplification
|-
|<math>\begin{align}
p\\
q\\
\therefore \overline{p \wedge q} \\
\end{align}</math>
| <math>((p) \wedge (q)) \rightarrow (p \wedge q)</math>
| Conjunction
|-
|<math>\begin{align}
p\\
p \rightarrow q\\
\therefore \overline{q \quad \quad \quad} \\
\end{align}</math>
| <math>((p \wedge (p \rightarrow q)) \rightarrow q</math>
| [[Modus ponens]]
|-
|<math>\begin{align}
\neg q\\
p \rightarrow q\\
\therefore \overline{\neg p \quad \quad \quad} \\
\end{align}</math>
| <math>((\neg q \wedge (p \rightarrow q)) \rightarrow \neg p</math>
| [[Modus tollens]]
|-
|<math>\begin{align}
p \rightarrow q\\
q \rightarrow r\\
\therefore \overline{p \rightarrow r} \\
\end{align}</math>
| <math>((p \rightarrow q) \wedge (q \rightarrow r)) \rightarrow (p \rightarrow r)</math>
| Hypothetical syllogism
|-
|<math>\begin{align}
p \vee q \\
\neg p \\
\therefore \overline{q \quad \quad \quad} \\
\end{align}</math>
| <math>((p \vee q) \wedge \neg p) \rightarrow q</math>
| Disjunctive syllogism
|-
|<math>\begin{align}
p \vee q \\
\neg p \vee r \\
\therefore \overline{q \vee r} \\
\end{align}</math>
| <math>((p \vee q) \wedge (\neg p \vee r)) \rightarrow (q \vee r)</math>
| Resolution
|}

All rules use the basic logic operators. A complete table of "logic operators" is shown by a [[truth table]], giving definitions of all the possible (16) truth functions of 2 [[Boolean algebra (logic)|boolean variables]] (''p'', ''q''):

{| class="wikitable" style="margin:1em auto 1em auto; text-align:center;"
|-
! ''p'' || ''q''
|
!  0  ||  1  ||  2  ||  3  ||  4  ||  5  ||  6  ||  7 
|
!|  8  ||  9  || 10 || 11 || 12 || 13 || 14 || 15
|-
! T || T
| || F || F || F || F || F || F || F || F || || T || T || T || T || T || T || T || T
|-
! T || F
| || F || F || F || F || T || T || T || T || || F || F || F || F || T || T || T || T
|-
! F || T
| || F || F || T || T || F || F || T || T || || F || F || T || T || F || F || T || T
|-
! F || F
| || F || T || F || T || F || T || F || T || || F || T || F || T || F || T || F || T
|}

where T = true and F = false, and, the columns are the logical operators: '''0''', false, [[Contradiction]]; '''1''', NOR, [[Logical NOR]]; '''2''', [[Converse nonimplication]]; '''3''', '''¬p''', [[Negation]]; '''4''', [[Material nonimplication]]; '''5''', '''¬q''', Negation; '''6''', XOR, [[Exclusive disjunction]]; '''7''', NAND, [[Logical NAND]]; '''8''', AND, [[Logical conjunction]]; '''9''', XNOR, [[If and only if]], [[Logical biconditional]]; '''10''', '''q''', [[Projection function]]; '''11''', if/then, [[Logical implication]]; '''12''', '''p''', Projection function; '''13''', then/if, [[Converse implication]]; '''14''', OR, [[Logical disjunction]]; '''15''', true, [[Tautology (logic)|Tautology]].

Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:
* The column-14 operator (OR), shows ''Addition rule'': when ''p''=T (the hypothesis selects the first two lines of the table), we see (at column-14) that ''p''∨''q''=T.
*: We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.
* The column-8 operator (AND), shows ''Simplification rule'': when ''p''∧''q''=T (first line of the table), we see that ''p''=T.
*: With this premise, we also conclude that ''q''=T, ''p''∨''q''=T, etc. as showed by columns 9-15.
* The column-11 operator (IF/THEN), shows ''Modus ponens rule'': when ''p''→''q''=T and ''p''=T only one line of the truth table (the first) satisfies these two conditions. On this line, ''q'' is also true. Therefore, whenever p → q is true and p is true, q must also be true.
Machines and well-trained people use this [[Lookup table|look at table approach]] to do basic inferences, and to check if other inferences (for the same premises) can be obtained.

===Example 1===
Let us consider the following assumptions: "If it rains today, then we will not go on a canoe today. If we do not go on a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for "therefore" is <math>\therefore</math>), if it rains today, we will go on a canoe trip tomorrow.
To make use of the rules of inference in the above table we let <math>p</math> be the proposition "If it rains today", <math>q</math> be " We will not go on a canoe today" and let <math>r</math> be "We will go on a canoe trip tomorrow". Then this argument is of the form:

<math>\begin{align}
p \rightarrow q\\
q \rightarrow r\\
\therefore \overline{p \rightarrow r} \\
\end{align}</math>

===Example 2===
Let us consider a more complex set of assumptions: "It is not sunny today and it is colder than yesterday". "We will go swimming only if it is sunny", "If we do not go swimming, then we will have a barbecue", and "If we will have a barbecue, then we will be home by sunset" lead to the conclusion "We will be home before sunset."
Proof by rules of inference: Let <math>p</math> be the proposition "It is sunny this today", <math>q</math> the proposition "It is colder than yesterday", <math>r</math> the proposition "We will go swimming", <math>s</math> the proposition "We will have a barbecue", and <math>t</math> the proposition "We will be home by sunset". Then the hypotheses become <math>\neg p \wedge q, r \rightarrow p, \neg r \rightarrow s</math> and <math>s \rightarrow t</math>. Using our intuition we conjecture that the conclusion might be <math>t</math>. Using the Rules of Inference table we can proof the conjecture easily:
{| class="wikitable"
|-
! Step
! Reason
|-
| 1.<math>\neg p \wedge q</math>
| Hypothesis
|-
| 2. <math>\neg p</math>
| Simplification using Step 1
|-
| 3. <math>r \rightarrow p</math>
| Hypothesis
|-
| 4. <math>\neg r</math>
| Modus tollens using Step 2 and 3
|-
| 5. <math>\neg r \rightarrow s</math>
| Hypothesis
|-
| 6. <math>s</math>
| Modus ponens using Step 4 and 5
|-
| 7. <math>s \rightarrow t</math>
| Hypothesis
|-
| 8. <math>t</math>
| Modus ponens using Step 6 and 7
|}

==References==
<references/>

{{Logic}}

{{DEFAULTSORT:List Of Rules Of Inference}}
[[Category:Rules of inference|*]]
[[Category:Mathematics-related lists|Rules of inference]]
[[Category:Philosophy-related lists|Rules of inference]]

[[de:Schlussregel]]
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Synchrotron radiation

2013-12-31T18:00:30Z

173.73.82.14: /* Frequency distribution of radiated energy */

{{About|a piecewise constant function|the unit step function|Heaviside step function}}

In [[mathematics]], a [[function (mathematics)|function]] on the [[real number]]s is called a '''step function''' (or '''staircase function''') if it can be written as a [[finite set|finite]] [[linear combination]] of [[indicator function]]s of [[interval (mathematics)|interval]]s. Informally speaking, a step function is a [[piecewise]] [[constant function]] having only finitely many pieces.
[[Image:StepFunctionExample.png|thumb|right|250px|Example of a step function (the red graph). This particular step function is [[Continuous_function#Directional_and_semi-continuity|right-continuous]].]]

==Definition and first consequences==
A function <math>f: \mathbb{R} \rightarrow \mathbb{R}</math> is called a '''step function''' if it can be written as {{Citation needed|date=September 2009}}

:<math>f(x) = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}(x)\,</math> for all real numbers <math>x</math>

where <math>n\ge 0,</math> <math>\alpha_i</math> are real numbers, <math>A_i</math> are intervals, and <math>\chi_A\,</math> (sometimes written as <math>1_A</math>) is the [[indicator function]] of <math>A</math>:

:<math>\chi_A(x) =
\begin{cases}
1 & \mbox{if } x \in A, \\
0 & \mbox{if } x \notin A. \\
\end{cases}
</math>

In this definition, the intervals <math>A_i</math> can be assumed to have the following two properties:

# The intervals are [[disjoint set|disjoint]], <math>A_i\cap A_j=\emptyset</math> for <math>i\ne j</math>
# The [[union (set theory)|union]] of the intervals is the entire real line, <math>\cup_{i=0}^n A_i=\mathbb R.</math>

Indeed, if that is not the case to start with, a different set of intervals can be picked for which these assumptions hold. For example, the step function

: <math>f = 4 \chi_{[-5, 1)} + 3 \chi_{(0, 6)}\,</math>

can be written as

: <math>f = 0\chi_{(-\infty, -5)} +4 \chi_{[-5, 0]} +7 \chi_{(0, 1)} + 3 \chi_{[1, 6)}+0\chi_{[6, \infty)}.\,</math>

==Examples==
[[Image:Dirac distribution CDF.svg|325px|thumb|The [[Heaviside step function]] is an often used step function.]]
* A [[constant function]] is a trivial example of a step function. Then there is only one interval, <math>A_0=\mathbb R.</math>
* The [[Heaviside step function|Heaviside function]] ''H''(''x'') is an important step function. It is the mathematical concept behind some test [[Signal (electronics)|signals]], such as those used to determine the [[step response]] of a [[dynamical system (definition)|dynamical system]].
[[File:Rectangular function.svg|thumb|The [[rectangular function]], the next simplest step function.]]
* The [[rectangular function]], the normalized [[boxcar function]], is the next simplest step function, and is used to model a unit pulse.

=== Non-examples ===
* The [[integer part]] function is not a step function according to the definition of this article, since it has an infinite number of intervals. However, some authors define step functions also with an infinite number of intervals.<ref>for example see: {{Cite book | author=Bachman, Narici, Beckenstein | title=Fourier and Wavelet Analysis | publisher=Springer, New York, 2000 | isbn=0-387-98899-8 | chapter =Example 7.2.2}}</ref>

==Properties==

* The sum and product of two step functions is again a step function. The product of a step function with a number is also a step function. As such, the step functions form an [[algebra over a field|algebra]] over the real numbers.
* A step function takes only a finite number of values. If the intervals <math>A_i,</math> <math>i=0, 1, \dots, n,</math> in the above definition of the step function are disjoint and their union is the real line, then <math>f(x)=\alpha_i\,</math> for all <math>x\in A_i.</math>
* The [[Lebesgue integral]] of a step function <math>\textstyle f = \sum\limits_{i=0}^n \alpha_i \chi_{A_i}\,</math> is <math>\textstyle \int \!f\,dx = \sum\limits_{i=0}^n \alpha_i \ell(A_i),\,</math> where <math>\ell(A)</math> is the length of the interval <math>A,</math> and it is assumed here that all intervals <math>A_i</math> have finite length. In fact, this equality (viewed as a definition) can be the first step in constructing the Lebesgue integral.<ref>{{Cite book | author=Weir, Alan J | authorlink= | coauthors= | title=Lebesgue integration and measure | date= | publisher=Cambridge University Press, 1973 | location= | isbn=0-521-09751-7 | unused_data=|chapter= 3}}</ref>

==See also==
*[[Simple function]]
*[[Piecewise defined function]]
*[[Sigmoid function]]
*[[Step detection]]

==References==
{{reflist}}

{{DEFAULTSORT:Step Function}}
[[Category:Special functions]]

Abraham–Lorentz force

2013-11-26T15:10:53Z

173.73.79.215: /* Derivation */ fixed inconsistencies with vector notation

{{about|the reflection principle in complex analysis|reflection principles of set theory|Reflection principle}}

In [[mathematics]], the '''Schwarz reflection principle''' is a way to extend the domain of definition of an [[analytic function]] of a [[complex variable]] ''F'', which is defined on the [[upper half-plane]] and has well-defined and [[real number]] boundary values on the [[real axis]]. In that case, the putative extension of ''F'' to the rest of the [[complex plane]] is

:<math>\overline{F(\bar{z})}</math>

or

:<math>F(\bar{z})=\overline{F(z)}.</math>

That is, we make the definition that agrees along the real axis.

The result proved by [[Hermann Schwarz|H. A. Schwarz]] is as follows. Suppose that ''F'' is a [[continuous function]] on the closed upper half plane <math>\left\{ z \in \mathbb{C}\ |\ \mathrm{Im}(z) \geq 0 \right\} </math>, [[holomorphic]] on the upper half plane <math>\left\{ z \in \mathbb{C}\ |\ \mathrm{Im}(z) > 0 \right\} </math>, which takes real values on the real axis. Then the extension formula given above is an [[analytic continuation]] to the whole complex plane.

In practice it would be better to have a theorem that allows ''F'' certain singularities, for example ''F'' a [[meromorphic function]]. To understand such extensions, one needs a proof method that can be tweaked. In fact [[Morera's theorem]] is well adapted to proving such statements. [[Contour integral]]s involving the extension of ''F'' clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results.

The principle also adapts to apply to [[harmonic function]]s.

==See also==
*[[Kelvin transform]]
*[[Method of image charges]]
==External links==
* {{springer|title=Riemann-Schwarz principle|id=p/r081990}}
*{{mathworld|SchwarzReflectionPrinciple}}

[[Category:Complex analysis]]
[[Category:Harmonic functions]]
[[Category:Theorems in complex analysis]]
[[Category:Mathematical principles]]

Subsonic and transonic wind tunnel

2011-12-15T21:53:19Z

173.73.22.167: /* Closed wind tunnel */

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