# Prove the equivalence of regular expressions

I have a question relating to regular expressions that I'm a bit confused about, If someone can help me out, that would be very much appreciated.

Suppose there exists regular expressions R, S and T.

Then is R*R* R*?

I know the way to prove this is by showing that L(R*) is a subset of L(R*R*) and vice versa, but I have no idea how to do that.. Just started learning about regular languages so I'm a bit confused. Can I please get any direction on how to do this?

• The languages denoted by RS and RT aren't the same. 101 is in RS but not in RT. You're on the right track, though. Welcome to the site! – Rick Decker Dec 3 '19 at 22:19

Start with $$R^*R^* = \left(\bigcup_{i=0}^\infty R^i\right) \left(\bigcup_{j=0}^\infty R^j\right) = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^i R^j = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j}.$$ As $$i,j$$ go over all pairs of non-negative integers, $$i+j$$ goes over all non-negative integers, hence $$R^*R^* = \bigcup_{i=0}^\infty \bigcup_{j=0}^\infty R^{i+j} = \bigcup_{k=0}^\infty R^k = R^*.$$