Transitive Relations - Revision history

Laouris: /* Examples of Transitive Relations */

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Examples of Transitive Relations

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Added examples

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			==ISM is based on Transitive Logic==
			[[Interpretive Structural Modeling\|ISM]] is based on the mathematical concept of Transitive Logic. This concept is used to reduce the number of comparisons (and thus time) needed to construct the MAP. Transitive Logic is also used to simplify the topology of the visual model. Transitive Logic states that for any 3 elements (A, B, C) with a given relation (�) when:

			A has the relation to B, (written A*B)<br>
			And B has the relation to C, (written B*C)<br>
			Then A has the relation to C, (written AC or AB*C)

			<u>For example:</u>
			If travel by car is faster than travel by bicycle<br>
			And travel by bicycle is faster than travel by walking<br>
			Then travel by car is faster than travel by walking

			Applying transitive logic also means that for any 3 elements (A, B, D) with a given relation, when:
			A has the relation to <br>
			And A does not have the relation to D<br>
			Then B does not have the relation to D<br>
			This means that B cannot have the relation to D or, using transitive logic, A would have to have the relation to D also

			<u>For example:</u>
			If travel by car is faster than travel by bicycle<br>
			And travel by car is not faster than travel by airplane<br>
			Then travel by bicycle is not faster than travel by airplane<br>


			[[ISM Software]] uses this type of logic to infer some of the relationships based on the answers given by the participants of a [[Structured Democratic Dialogue]]. The number of inferred answers in an ISM session varies with the situation, but it is often in the order of 70%. This represents considerable time-saving.

	=== Examples of Transitive Relations ===		=== Examples of Transitive Relations ===

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	*		*
	[[Category:Linear Algebra]]		[[Category:Linear Algebra]]
			[[Category: ISM Terminology]]

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation <span style="font-size:~~110~~%">''R'' </span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:130%">''R'' </span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

Laouris at 14:47, 9 January 2022

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation <span style="font-size:~~150~~%">''R'' "</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:110%">''R'' </span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

Laouris at 14:47, 9 January 2022

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation <span style="font-size:150%">R~~; font-style=italic~~''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:150%">''R'' "</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

Laouris at 14:46, 9 January 2022

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation <span style="font-size:150%">R; ~~italics~~''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:150%">R; font-style=italic''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

Laouris at 14:45, 9 January 2022

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation <span style="font-size:~~200~~%">R''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:150%">R; italics''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

Laouris at 14:44, 9 January 2022

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	===Mathematical Definition===		===Mathematical Definition===
	A binary relation ~~{{huge\|~~R}} defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.		A binary relation <span style="font-size:200%">R''</span> defined on a set S is said to be transitive if, for any elements A, B, and C in the set S, given that A R B and B R C, it necessarily follows that A R C.

	~~<blockquote style="background-color: lightgrey; border: solid thin grey; padding-left: 16px; padding-right: 6px;"> Your quote here — Annon </blockquote>~~

	~~<span style="font-size:88%">German: ''Der Lange-Annoying-Name-der-Dinge''</span>~~

Laouris at 14:43, 9 January 2022

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	<blockquote style="background-color: lightgrey; border: solid thin grey; padding-left: 16px; padding-right: 6px;"> Your quote here — Annon </blockquote>		<blockquote style="background-color: lightgrey; border: solid thin grey; padding-left: 16px; padding-right: 6px;"> Your quote here — Annon </blockquote>

			<span style="font-size:88%">German: ''Der Lange-Annoying-Name-der-Dinge''</span>


	=== Examples of Transitive Relations ===		=== Examples of Transitive Relations ===