Mathematical principles of reinforcement - Revision history

en>Yobot: /* References */WP:CHECKWIKI error #10 + general fixes using AWB (8888)

2013-02-08T11:45:24Z

References: WP:CHECKWIKI error #10 + general fixes using AWB (8888)

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Moisture is extremely important, eѕpecially given that we are fundamentally made of drinking water. You everyday drinking Һabits objectiѵe ought to be at least 8-10 cuρs.<br><br>Using the economic system in the shape it is in now, substantial numbers of men and women want to taкe adѵantage of coupon codes. On the present "Excessive Couponing," everyone is paying out fundamentally notҺing for cɑrtfuls of household goods. They may be getting deeply savings, but it is mainly on processeԁ garЬage. Promotion deals typically are most oftеn for high-fat, substantial-calories goods.<br><br>As уou may now know, obtaining apprօpriate nutrition iѕ absolutely not too hard. Examine the issues linked to nutrition. You may be rewаrded with much better health. Thеse guidance will allow you to crеate the correct alternatives.		In [[constructivism (mathematics)\|constructive mathematics]], a '''pseudo-order''' is a constructive generalisation of a [[linear order]] to the continuous case. The usual [[trichotomy (mathematics)\|trichotomy law]] does not hold in the constructive continuum because of its [[indecomposability]], so this condition is weakened.

			A pseudo-order is a [[binary relation]] satisfying the following conditions:
			# It is not possible for two elements to each be less than the other. That is, <math>\forall x,y: \neg\;(x < y \;\wedge\; y < x)</math>.
			# For all {{mvar\|x}}, {{mvar\|y}}, and {{mvar\|z}}, if {{math\|''x'' < ''y''}} then either {{math\|''x'' < ''z''}} or {{math\|''z'' < ''y''}}. That is, <math>\forall x,y,z: x < y \;\to\; (x < z \;\vee\; z < y)</math>.
			# Every two elements for which neither one is less than the other must be equal. That is, <math>\forall x,y: \neg\;(x < y \;\vee\; y < x) \;\to\; x = y</math>
			This first condition is simply [[antisymmetric relation\|antisymmetry]]. It follows from the first two conditions that a pseudo-order is [[transitive relation\|transitive]]. The second condition is often called ''co-transitivity'' or ''comparison'' and is the constructive substitute for trichotomy. In general, given two elements of a pseudo-ordered set, it is not always the case that either one is less than the other or else they are equal, but given any nontrivial interval, any element is either above the lower bound, or below the upper bound.

			The third condition is often taken as the definition of equality. The natural [[apartness relation]] on a pseudo-ordered set is given by
			: <math>x \# y \;\leftrightarrow\; x < y \;\vee\; y < x</math>
			and equality is defined by the negation of apartness.

			The negation of the pseudo-order is a [[partial order]] which is close to a [[total order]]: if ''x'' ≤ ''y'' is defined as the negation of ''y'' < ''x'', then we have
			: <math> \neg\;(\neg\;(x \le y) \;\wedge\; \neg\;(y \le x)) .</math>
			Using [[classical logic]] one would then conclude that ''x'' ≤ ''y'' or ''y'' ≤ ''x'', so it would be a total order. However, this inference is not valid in the constructive case.

			The prototypical pseudo-order is that of the real numbers: one real number is less than another if [[there exists]] (one can construct) a rational number greater than the former and less than the latter. In other words, ''x'' < ''y'' if there exists a rational number ''z'' such that ''x'' < ''z'' < ''y''.

			==References==
			* [[Arend Heyting]] (1966) Intuitionism: An introduction. Second revised edition North-Holland Publishing Co., Amsterdam.
			http://books.google.com/books/about/Intuitionism.html?id=4rhLAAAAMAAJ
			<references />

			[[Category:Constructivism (mathematics)]]
			[[Category:Order theory]]


			{{mathlogic-stub}}

en>Robin S: "that describe [the most fundamental aspects of behavior" -> "that attempt to describe" - the research on this subject so far has been limited

2011-07-20T23:43:54Z

"that describe [the most fundamental aspects of behavior" -> "that attempt to describe" - the research on this subject so far has been limited

New page

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