Give a regular expression for the language L1 of strings over alphabet Σ = {a, b, c} that have a pair of identical symbols separated by a substring whose length is a multiple of 3. For example, abcba, bb, cbabcb ∈ L1 but ε, a, bcbcab /∈ L1. Note that cbabcb is in the language two different ways. Briefly justify your answer.

Explanation:

(a(aaa+aab+aac+aba+abb+abc+aca+acb+acc+baa+bab+bac+bba+bbb+bbc+bca+bcb+bcc+caa+cab+cac+cba+cbb+cbc+cca+ccb+ccc)^{*}a)\,\,+\,\,(b(aaa+aab+aac+aba+abb+abc+aca+acb+acc+baa+bab+bac+bba+bbb+bbc+bca+bcb+bcc+caa+cab+cac+cba+cbb+cbc+cca+ccb+ccc)^{*}b)\,\,+\,\,(c(aaa+aab+aac+aba+abb+abc+aca+acb+acc+baa+bab+bac+bba+bbb+bbc+bca+bcb+bcc+caa+cab+cac+cba+cbb+cbc+cca+ccb+ccc)^{*}c)

The strings generated are as follows:

'a' separated from 'a' by any string which is a multiple of 3

or

'b' separated from 'b' by any string which is a multiple of 3

or

'c' separated from 'c' by any string which is a multiple of 3.

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why wouldnt you just google the words bud? not that hard to