# 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.