# DFA of alternate 0’s and 1’s

• Difficulty Level : Medium
• Last Updated : 09 Sep, 2020

Regular Expression can be anything, from a terminal symbol, ∅, to union of two regular expressions (R1 + R2), their concatenation (R1R2) or its closure R1* as well.

Examples of Regular Expression :

1. Regular expression of set of all strings of 0’s and 1’s starting with two zeros :
`00(0+1)*`
2. 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)*`
3. 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 –

1. ∈ (no input, 0 and 1)
2. 010101….. (string that starts with 0 followed by 1 and so on).
3. 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

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