Question

if $w \in (a+b)^*$ satisfies $abw = wab$, then $\text{length}(w)$ is ?

Answer 1

abw = wab

one of solution is, w = (ab)*  

[w can be any string ∈ (ab)* ]

So length of W must be even for sure.
(But all even length string ∈{a,b}* may not satisfy abW = Wab)

Comments

if only some of strings can satisfy then how we can say that it is even length?

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Relate following : S = {2,5} Cardinality of set S is 2. Cardinality of set S is Even. Both statement are true.

Answer 2

|w| must be even