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There are some threads that discuss it but I haven't came across an inductive one yet. All of them involve creating a finite automaton which I would like to avoid (as per my professors requests).

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You can use regular expressions to easily solve this:

$$(L_1\cup L_2)^r=L_1^r\cup L_2^r$$ $$(L_1\cap L_2)^r=L_1^r\cap L_2^r$$ $$(L_1L_2)^r=L_2^rL_1^r$$ $$(L^*)^r=(L^r)^*$$

You can use these equalities when proving by induction over the number of operators in the regular expression.

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