Question

Define a transitive 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 transitive relation as a relation defined on a single set $ A $ as $ R:A\to A $ and if it has ordered pairs of the form $ \left( a,b \right),\left( b,c \right) $ which ensures the order pair $ \left( a,c \right) $ to be in $ R $ where $ a,b,c $ are distinct elements 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 $ . The ordered pair $ \left( a,b \right) $ signifies that $ a $ is related to $ b $ that is $ aRb $ .
Let $ R $ be binary relation on the set $ A $ that is $ R:A\to A $ . We call $ R $ a transitive relation for three distinct elements $ a,b,c $ of $ A $ if $ R $ relates $ a $ to $ b $ and $ b $ to $ c $ then $ R $ also relates $ a $ to $ c $ . It means
\[\begin{align}
  & aRb,bRc\Rightarrow aRc \\
 & \left( a,b \right)\in R,\left( b,c \right)\in R\Rightarrow \left( a,c \right)\in R \\
\end{align}\]
The word transitive comes from the understanding that the relation $ R $ transitions from $ a $ to $ c $ . Let us check some examples of transitive relation. The inequalities relations defined over real number set less than, less than equal to, greater than, greater than equal to are transitive relations because for all $ x,y,z\in \mathsf{\mathbb{R}} $
\[\begin{align}
  & x < y,x < z\Rightarrow x < z \\
 & x\le y,y\le z\Rightarrow x\le z \\
 & x > y , y > y\Rightarrow x > z \\
 & x\ge y,y\ge z\Rightarrow x\ge z \\
\end{align}\]
 The relation ‘is a subset of’ among sets defined on universal set $ U $ is a transitive relation because for all $ A,B,C\in U $
\[A\subseteq B,B\subseteq C\Rightarrow A\subseteq C\]
The relation ‘divides’ or ‘exactly divides ’ among integers is a transitive relation because for all $ a,b,c\in \mathsf{\mathbb{Z}} $
\[b|a,c|b\Rightarrow c|a\]

Note:
We note that the relation $ R $ is called intransitive if $ \left( a,b \right)\in R,\left( b,c \right)\in R $ the form some $ a,b,c $ we have $ \left( a,c \right)\in R $ and $ R $ is called anti-transitive if $ \left( a,b \right)\in R,\left( b,c \right)\in R\Rightarrow \left( a,c \right)\notin R $ . For example $ \left( a,b \right)\in R $ if $ ab $ is an even number is intransitive but not anti-transitive. The relation $ \left( a,b \right)\in R $ if $ a $ is an even number and $ b $ is an odd number is both intransitive but not anti-transitive. If relation is reflexive, symmetric and transitive it is called equivalence relation.