Given $\sum =\{a,b\}$ , give some example of strings in, and not in these sets where: $\{w\mid: ww=www\}$

Question

For a string not in $\{w\mid: ww=www\}$ one can simply choose $w=a$

But, the only way I can see a string being in $\{w\mid: ww=www\}$ is if $w=\lambda$ (the empty string), but $\lambda$ is not part of the alphabet. Is this wrong?

Answer 1

Although $\lambda$ is not in the alphabet, it is still a string, and in fact it is the only string in that language. Since $ww\lambda$=$www$, it follows that $w=\lambda$.