self doubt

Question

if A is regular,

L= HALF(A)= { x | for some y, |x|= |y| and xy ϵ A} is regular

 i want to ask if i know DFA for A, how to construct DFA for L ??

Comments

https://gateoverflow.in/93981/half-l

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the explanation given there is just initutative, i have also solved it on intution based,

if i give you regular expression for 'A' ,can you write regular expression for L ??

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@ joshi_nitish 

Read this :

http://www-bcf.usc.edu/~breichar/teaching/2011cs360/half(L)example.pdf