Define a relation $$ on the set $S$ of all strings of letters: two strings are related if you can get one from the other by reversing one pair of adjacent letters. For example, cow $$ ocw but cowt woc.

Consider all the strings you can form with the letters $c,a,$ and $t($there are six$).$ Draw the graph whose nodes are these six strings and whose edges represent the $$ relation. Should this be a directed or an undirected graph?

Find an Euler path in the graph you made in Exercise $1.$

Consider the graph formed by the $$ relation on the set of all the strings you can form from the letters I, ry$,n,$ and $x.$ Does this graph have an Euler path? Why or why not?

Does the graph of the $$ relation on the set of all strings formed from the letters I,$e,0,p,a,r,$ and d have an Euler path? Why or why not?

The $"-=$ mod $3"$ relation is an equivalence relation on the set $\{1,2,3,4,5,6,7\}.$ List the equivalence classes.

In class we showed that $"-=$ mod $n"$ relation on $Z$ is transitive. Show that it is also symmetric and reflexive.