Question

Define a reflexive relation. \[\]

Answer 1

Hint: We recall a binary relation between two sets A and B as the non-empty subset of their Cartesian product $ A\times B $. We recall a reflexive relation as a relation defined on a single set $ A $ as $ R:A\to A $ and has ordered pairs of the form $ \left( a, a \right) $ where $ a $ is an element of $ A $ . \[\]

Complete step by step answer:
We know that a binary relation $ R $ over sets $ A $ and $ B $ is a non-empty subset of Cartesian product $ A\times B $ which means $ R $ is a set of ordered pairs $ \left( a,b \right) $ such that $ a $ is from $ A $ and $ b $ is from $ B $ .\[\]
We know that we call a binary relation $ R $, a reflexive relation over the set $ A $ if the relation relates every element of $ A $ with itself. If $ R $ is a reflexive relation defined onset $ A\times A $ then we have $ aRa $ for all $ a\in A $ . If we define the set $ A $ as $ A=\{w,x,y,z\} $ then one of the reflexive relations is given as;
\[R=\left\{ \left( w,w \right),\left( x,x \right),\left( y,y \right),\left( z,z \right),\left( w,x \right) \right\}\]
Let us take examples of reflexive relations. We take the natural number set $ A=N $ and define the relation as “is equal to”. Since every number is equal to itself $ \left( 1=1,2=2,3=3,... \right) $ every natural number is related to itself by the relation “is equal to”. So the relation is given by;
\[R=\left\{ \left( a,b \right):a=b,a\in N,b\in N \right\}\]
Hence the relation “is equal to” is reflexive relation. We can other examples of reflexive relation as “greater then equal to”, “ is a subset of, “divides” etc. \[\]

Note:
 We must be careful of the different between identity relation and reflexive relation. We can define an identity relation over the set $ A $ as $ I=\left\{ \left( a,a \right):a\in A \right\} $ . The identity relation is unique but reflexive relation may not be unique. Every identity relation is reflexive relation $ \left( I\subseteq R \right) $ but the opposite may not be true. For example the identity relation of “is equal to” in natural number is $ I=\left\{ \left( 1,1 \right),\left( 2,2 \right),\left( 3,3 \right)... \right\} $ but one of the reflexive relation is $ R=\left\{ \left( 1,2 \right),\left( 2,1 \right),\left( 1,1 \right),\left( 2,2 \right),..., \right\} $ . If no element is related itself in a set by relation $ R $ , then we call $ R $ an anti-reflexive relation.