Suppose you have three states in an DFA $q_{1}$, $q_{2}$ and $q_{3}$, all non-accepting. Suppose $a$ and $b$ are two symbols, such that:$\delta(q_{1},a)=q_{2}\\ \delta(q_{2},a)=q_{1}$and$\delta(q_{1},b)=q_{3}\\ \delta(q_{2},b)=q_{3}$Are the states $q_{1}$ and $q_{2}$ considered equivalent?
Are two states equivalent in a DFA if on the same input the states transition to each other?
finite automata;equality
Not necessarily.  If there is another symbol in the alphabet, $c$, such that $q_1$ transitions to $q_4$ on $c$ but $q_2$ transitions to $q_5$ then they are clearly not equivalent.