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

Revision as of 03:30, 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 $x \leq z$ .

Another example is given sets $A, B, C$ we can write $A \subseteq B \subseteq C$ or $A \supseteq B \supseteq C$ .

In fact, the relation that is used does not have to be the same in both places. We might write $p \in B \subseteq U$ to mean " $p \in B$ and $B \subseteq U$ ".

@@ Line 2: / Line 2: @@
 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>.
+Another example is given sets <math>A,B,C</math> we can write <math>A \subseteq B \subseteq C</math> or <math>A \supseteq B \supseteq C</math>.
+In fact, the relation that is used does not have to be the same in both places. We might write <math>p \in B \subseteq U</math> to mean "<math>p \in B</math> and <math>B \subseteq U</math>".