# Let u and v be two strings. What about the reverse order of their concatenaited string?

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.

• Have you tried a few strings? For example, $u=0$, $v=1$? – John L. Feb 17 '19 at 10:55
• @Apass.Jack I have tried to prove it. But I couldn't.So I thought maybe they are not equal. – siaVash Feb 17 '19 at 11:03
• Let me rephrase my suggestion. Have you checked that equality with a few concrete instances? For example, what are $(u.v)^R$ and $u^R.v^R$ when $u$ is the string 0 and $v$ is the string 1? – John L. Feb 17 '19 at 11:06

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 $$(u\cdot v)^R=v^R\cdot u^R$$.

• I have to use induction to prove it? – siaVash Feb 17 '19 at 14:05
• Yes, mathematically speaking. It becomes clear and obvious once you have played with several examples. – John L. Feb 17 '19 at 14:08
• Inequality "more often than not": to be precise we have $(uv)^R = u^Rv^R$ precisely when $u$ and $v$ are powers of the same word – Hendrik Jan Feb 17 '19 at 16:44