Regular expression for containing 010 as a subword

Question

I am studying for a test in computer science, and am encountering difficulties with regular expression. Here is example of a question I don't understand.

I managed to solve the following question:

Give a regular expression for the language of all words over $\{a,b\}$ which contains one of $aa,bb$ as a subword.

My solution: $$(a ∪ b)^*(aa ∪ bb)^*(a ∪ b)^*.$$

I am having trouble with the following question:

Give a regular expression for the language of all words over $\{0,1\}$ which contains $010$ as a subword.

L1 does not look right. All the words in L1 are such that "aa" or "bb" occurs as a substring, but not all words that satisfy the above property are in L1. — Steven, Nov 08 '19 at 22:00

score 1 · Answer 1 · edited Nov 10 '19 at 09:35

In the first one, L1 = (a+b)*(aa+bb)(a+b)* not (a+b)(aa+bb)(a+b)* because in the second one, it is possible only to have one character of a or b and then we have to find the occurrence we need. Your L1 denies any word that does start with (aa+bb). It also forces all the words to start with a or b.

By the way, I use + for | and ∪.

So for the second one, it is in the same manner:

L1 = (0+1)*(010)(0+1)*

010 might occurs in the beginning, the middle or the end of the word.

Regular expression for containing 010 as a subword

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