7. Consider the language L = {w | w {a, b}* and |w|a = 3 and w has odd length }.

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A language is called regular if its the language of a DFA. The following algorithm describes how to construct the intersection of two regular languages by constructing a new DFA from the DFAs of the input languages:

Suppose L1 and L2 are regular languages on the same alphabet. Then by regularity we know that there must exist two DFAs M1 = (Q1, , 1, q1, F1) and M2 = (Q2, , 2, q2, F2) such that L(M1) = L1 and L(M2) = L2. We construct a DFA, M = (Q, , , q, F), where L(M) = L1 L2 as follows:

Q = Q1 Q2

=

((r1, r2), a) = (1(r1, a), (r2, a))

q = (q1, q2)

F = F1 F2

Consider the language L = {w | w {a, b} and |w|a = 3 and w has odd length }.

The language L is an intersection of two regular languages.

Complete the following:

i. Identify the two regular languages L1 and L2 such that L = L1 L2, and define DFAs for both.

ii. Using the solution to i and the intersection construction provided above, define the DFA that recognizes L1 L2 = L. DFA diagram required.

Any help is greatly appreciated, this homework was sprung on us late with little to no notice, and I have 3 other assignments due tomorrow. This question has stumped me.

Thank you so much for your assistance.