Intersection process of two DFAs

Prerequisite – Design a Finite automata

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 of 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.