# What is the formal definition of epsilon or ε-closure?

ε stand to epsilon or null value in NFA. ε-closures of state qi is the set of states including qi where qi can reach by any number of ε – moves of the given NFA.

## ε-closures of qi :-

ε-closures of a state qi, includes qi.

– Set of states reachable from qi on ε – moves.

– Set of states reachable from existing states in ε-closures, using ε-move, and so on.

ε-closures of various states in above figure are given below:

ε-closures of q0 = {q0, q1, q2, q4, q7}

There are ε-moves from q0 to q1 ,
q0 to q7 ,
q1 to q2 ,
q1 to q4 .

ε-closures of q1 = {q1, q2, q4}

There is no ε-moves from q1 to q4 , q1 to q2 .

ε-closures of q2 = {q2}

There are ε-moves from q3 to q6 ,
q6 to q7 ,
q6 to q1 ,
q1 to q2 ,
q1 to q4.

ε-closures of q4 = {q4}

There is no ε-move from q4 .

ε-closures of q5 = {q5, q6, q7, q1, q2, q4}

There are ε-moves from q5 to q6 ,
q6 to q7 ,
q6 to q1 ,
q1 to q2 ,
q1 to q4.

ε-closures of q6 = {q7, q1, q2, q4}

There are ε-moves from q6 to q7 ,
q6 to q1 ,
q1 to q2 ,
q1 to q4.

ε-closures of q7 = {q7}

There is no ε-moves from q7 .

ε-closures of various states are summarized below :