problems

relations

1. Find a relation \(R\) on the set of English words \(W\) with each of the profiles given below: 

   1. irreflexive, asymmetric, and transitive on \(W\).

   2. reflexive and euclidean on \(W\).

   3. irreflexive, symmetric, and intransitive on \(W\). 

   Please justify your answers.  

2. Draw a diagram for a finite relation \(R\) on a set \(W\) with each of the profiles given below:

   1. reflexive, symmetric, and non-transitive on \(W\).

   2. non-reflexive, symmetric, and intransitive on \(W\).

   3. irreflexive, symmetric, and transitive on \(W\).

   4. euclidean, connected, and non-reflexive on \(W\).

   Please justify your answers. 

3. Justify each of the claims given below: 

   1. A symmetric relation \(R\) on a set \(A\) is transitive on \(A\) if, and only if, \(R\) is euclidean on \(A\).

   2. If a relation \(R\) is both reflexive and euclidean on a set \(A\), then \(R\) is symmetric on \(A\).

   3. \(R\) is an equivalence relation on a set \(A\) if, and only if, \(R\) is reflexive and euclidean on \(A\). 

4. True or false? If true, please provide an argument. If false, provide a counterexample.

   1. The empty set \(\emptyset\) is a binary relation on any set \(A\).

   2. If a binary relation \(R\) on a set \(A\) is reflexive, symmetric, and connected on \(A\), then \(R\) is euclidean on \(A\).

   3. If a binary relation \(R\) on a set \(A\) is euclidean and connected on \(A\), then \(R\) is either reflexive or symmetric on \(A\).