How to define an automata for zig zag concatenation? [duplicate]


I have two DFAs one for language A and one for language B. I’m asked to make an FDA that is the zig-zag concatenation of letters of A and letters of B. This is described by the following: {w: w = $ a_1 b_1$ $ a_k b_k$ and $ a_1…a_k \in A$ and $ b_1 … b_k \in B$ }. With $ 1 \leq i \leq k$ and $ a_i \in \Sigma$ and $ b_i \in \Sigma$

This automata should be described as a 5-tuple {Q, $ \Sigma$ ,$ \delta$ ,$ q_0$ ,$ F$ }.

I simply do not know how I would go about defining the total function $ \delta$ .

This is what I tried:

$ Q = Q_A \cap Q_B$ // We only want the words that contains both a and b

$ F = F_A \cap F_B$ // The accepted states should contain both letter from a and from b.

$ q_0 = q_A$ //because the word starts with a letter from a.

$ \Sigma$ // in this problem we aren’t interested in defining the alphabet we just leave the symbol as is in the 5-tuple.