let $u$ and $v$ be two strings. Is $(u.v)^R$ equals to $u^R.v^R$?
Note: The $R$ notation means reverse order and the $.(dot)$ notation means concatenation.
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No, $(u\cdot v)^R\not=u^R\cdot v^R$ more often than not.
For example, if $u$ is the string $race$ and $v$ is the string $car$. Then $$(u\cdot v)^R=(racecar)^R=racecar$$ while $$u^R\cdot v^R=(race)^R(car)^R=ecarrac.$$
Here are a few related exercises. All variables stand for strings.
Exercise 1. If $u$ or $v$ is the empty word, then $(u\cdot v)^R=u^R\cdot v^R$.
Exercise 2. If $u$ or $v$ are words of length 1 such that $(u\cdot v)^R=u^R\cdot v^R$. Then $u=v$.
Exercise 3. If $u=u^R$ and $v=v^R$, can we guarantee $(u\cdot v)^R=u^R\cdot v^R$?
Exercise 4. Prove that $(u\cdot v)^R=v^R\cdot u^R$.