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### What is the regular languages that are equivalent to the given regular language: (0 + 1)*1(0 + 1)*

## Question

I am learning automata theory. I have answered the question wrongly three times, and I don't know the right answers. Please help me.

The target language is `(0+1)*1(0+1)*`

. The choices are:

```
A) (01+11)*(0+1)*
B) (0+1)*(10+11+1)(0+1)*
C) (1+0)*1(1+0)*
D) (0+1)*(0+1)(0+1)*
```

Multiple choices are allowed, if there's more than one equivalent language.

## Best answer

Consider your options, and either factor them into a form that looks like your target, or find a word that is in one language but not the other:

Target: `(0+1)*1(0+1)*`

, this is the language of all words over `{0,1}`

containing at least one `1`

.

A) `(01+11)*(0+1)*`

, this does not include at least one `1`

. For example, the empty string is in A but not in the target.

B) `(0+1)*(10+11+1)(0+1)*`

, looking specifically at the `10+11+1`

part, which can be factored into `1(0+1+ε)`

. `(0+1+ε)(0+1)*`

is not different from `(0+1)*`

.

C) `(1+0)*1(1+0)*`

, this is just the target with `0+1`

replaced with `1+0`

. The union operation `+`

is commutative.

D) `(0+1)*(0+1)(0+1)*`

, this language includes the word `0`

which is not in the target.