Question: For languages A and B over , let the perfect shuffle be the language { w | w = a 1 b 1 a 2

For languages A and B over

,

let the perfect shuffle be the language

{w | w = a_{1} b_{1} a_{2} b_{2} d o t s a_{k} b_{k},

where

a_{1} d o t s a_{k}

inA and

b_{1} d o t s b_{k}

inB

}

Suppose the string "rose" is in the language A and "text" is in the language B

,

then the string "rtoesxet" is in the perfect shuffle of A and B

.

Show that the class of regular languages is closed under perfect shuffle: If

A

is a regular language and

B

is a regular language, then the perfect shuffle of A and

B

is a regular language. Hint: How can you use the DFAs for both languages and the cross product construction to construct an FA for the perfect shuffle?

For languages A and B over , let the perfect shuffle

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