Intersection Process of Two DFAs

Introduction

• Let's understand the intersection of two DFA with an example. 
• Designing a DFA for the set of string over {0, 1} such that it ends with 01 and has even number 0f 1's. 

There two desired language will be formed:

L1= {01, 001, 101, 0101, 1001, 1101, ....} 

L2= {11, 011, 101, 110, 0011, 1100, .....}

L = L₁ and L₂ = L₁ ∩ L₂

State Transition Diagram for the language L₁ :
This is a DFA for language L₁

It accepts all the string that accept 01 at end. 

State Transition Diagram for the language L2 : 
This is a DFA for language L₂ 

It accepts all the string that accept with even number of 1's.
State Transition Diagram of L₁ ∩ L₂ :
Intersection of L₁ and L₂ can be explained by language that a string over {0, 1} accept such that it ends with 01 and has even number of 1's.

L = L1 ∩ L2

= {1001, 0101, 01001, 10001, ....}

Thus as we see that L1 and L2 have been combined through intersection process and this final DFA accept all the language that has even number of 1's and is ending with 01.