User:IssaRice/Stringing together relations and binary operations: Difference between revisions

Revision as of 03:28, 3 August 2019

If $R$ is a relation on a set $X$ , and $x, y, z$ are elements of $X$ , we sometimes write $x R y R z$ as an abbreviation of " $x R y$ and $y R z$ . This makes sense especially when $R$ is a transitive relation, because in that case we also have $x R z$ , which is suggested by the notation " $x R y R z$ ".

For instance, if we have three real numbers $x, y, z$ and the relation $\leq$ , then $x \leq y \leq z$ means that $x \leq y$ and $y \leq z$ . Since the relation is transitive, we also have $x \leq z$ .

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	If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \~~mathrel~~{R} y \~~mathrel~~{R} z</math> as an abbreviation of "<math>x \~~mathrel~~{R} y</math> and <math>y \~~mathrel~~{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \~~mathrel~~{R} z</math>, which is suggested by the notation "<math>x \~~mathrel~~{R} y \~~mathrel~~{R} z</math>".		If <math>R</math> is a relation on a set <math>X</math>, and <math>x,y,z</math> are elements of <math>X</math>, we sometimes write <math>x \mathbin{R} y \mathbin{R} z</math> as an abbreviation of "<math>x \mathbin{R} y</math> and <math>y \mathbin{R} z</math>. This makes sense especially when <math>R</math> is a transitive relation, because in that case we also have <math>x \mathbin{R} z</math>, which is suggested by the notation "<math>x \mathbin{R} y \mathbin{R} z</math>".

	For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.		For instance, if we have three real numbers <math>x,y,z</math> and the relation <math>\leq</math>, then <math>x \leq y \leq z</math> means that <math>x \leq y</math> and <math>y \leq z</math>. Since the relation is transitive, we also have <math>x \leq z</math>.