# DFA of alternate 0’s and 1’s

**Regular Expression **can be anything, from a terminal symbol, ∅, to union of two regular expressions (R_{1} + R_{2}), their concatenation (R_{1}R_{2}) or its closure R_{1}* as well.

**Examples of Regular Expression :**

- Regular expression of set of all strings of 0’s and 1’s starting with two zeros :
00(0+1)*

- Regular expression of set of all strings of 0’s and 1’s having even number of 0’s followed by odd numbers of 1’s :
(00)*1(11)*

- Regular expression of set of all strings of 0’s and 1’s containing at least one 0 and at least two 1’s :
00*11(0+1)* + 0111*(0+1)*

Strings that will be acceptable by regular expression with alternate 0’s and 1’s –

- ∈ (no input, 0 and 1)
- 010101….. (string that starts with 0 followed by 1 and so on).
- 101010….. (string that starts with 1 followed by 0 and so on).

Now, a regular expression for set of all strings consisting of alternate 0’s and 1’s would be **(01)***, where it can accept ∈, 01, 0101, 010101…..etc but this restricts the string as it can always begin with 0 only.

Again, the expression **(10)*** will accept ∈, 10, 1010, 101010….etc but this too restricts string as it can always begin with 1 only.

So, we introduce** 1(01)*** and **0(10)*** to meet gap in respective cases.

While 1(01)* breaks the restriction of string starting with 0, 0(10)* breaks same for string starting with 1.

So, the final expression is –

(01)* + (10)* + 0(10)* + 1(01)*

**Figure –**Finite Automata of Regular Expression for alternate 0’s and 1’s