relations - NOTES ON RELATIONS These notes introduce the...

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Unformatted text preview: NOTES ON RELATIONS These notes introduce the notion of a relation and, more importantly, the notion of an equivalence relation. Definition 1. A relation R between two sets A and B is a subset of the Cartesian product A B . We say that a A and b B are related if ( a,b ) R and denote this by writing aRb . By a relation on a set A we mean a relation between A and itself. This definition may seem to be of little use, since we are just defining a relation as a subset of a Cartesian product with no additional requirements. The point is that it gives us a set-theoretic and precise way of defining what it means for two objects to be related (in whatever sense we are working with) to each other. Here is a simple example. Let A be the set of all people in the world and define a relation on A by R := { ( a,b ) A A | a is the father of b } . This set completely encodes the relation between father and child in a precise way, which is what we would want if we wanted to do some mathematics with this (which we do not). The most important types of relations are those which are known as equivalence relations : Definition 2. A relation on a set A is called an equivalence relation if it satisfies the following properties: The relation is reflexive , meaning that aRa for all a A . The relation is symmetric , meaning that aRb implies bRa . The relation is transitive , meaning that if aRb and bRc , then aRc . Equivalence relations are commonly denoted by , so instead of writing aRb we write a b and say that a and b are equivalent....
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relations - NOTES ON RELATIONS These notes introduce the...

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