Theory of Computation problem: Proving that L = L(A)

[s3a]

Hello to everyone that's reading this. :)

1. Homework Statement
(Theory of Computation) proof problem about proving that L = L(A) by proving that L ⊆ L(A) and that L(A) ⊆ L:
https://www.docdroid.net/du7lLvh/theproblemanditssolution.pdf

Homework Equations

• Mutual simple induction
• Formal usage of transition function of DFAs (meaning knowing how to use the delta notation)
• Potentially, proof by contrapositive
• Knowing that A = B can be proven by proving A ⊆ B and B ⊆ A.

The Attempt at a Solution

My question is: does the "δ̂(p, w) = p ⇒ w has even number of 1's" from second part (L ⊆ L(A)) need the first part (L(A) ⊆ L) to already be shown, or is it completely independent, such that the second part (L ⊆ L(A)) could have been cut and pasted, verbatim, before the first part (L(A) ⊆ L)?

Any input would be GREATLY appreciated!

 

[Mark44]

Hello to everyone that's reading this. :)

1. Homework Statement
(Theory of Computation) proof problem about proving that L = L(A) by proving that L ⊆ L(A) and that L(A) ⊆ L:
https://www.docdroid.net/du7lLvh/theproblemanditssolution.pdf

Homework Equations

• Mutual simple induction
• Formal usage of transition function of DFAs (meaning knowing how to use the delta notation)
• Potentially, proof by contrapositive
• Knowing that A = B can be proven by proving A ⊆ B and B ⊆ A.

The Attempt at a Solution

My question is: does the "δ̂(p, w) = p ⇒ w has even number of 1's" from second part (L ⊆ L(A)) need the first part (L(A) ⊆ L) to already be shown, or is it completely independent, such that the second part (L ⊆ L(A)) could have been cut and pasted, verbatim, before the first part (L(A) ⊆ L)?

Any input would be GREATLY appreciated!

Possibly the reason no one has replied is that there is not enough information here.
In the linked document it says

Let $L = \{w \in \{0, 1\}* : \text{w has an odd number of 1's} \}$, and let A be the DFA with tabular representation ...

1. Does the notation {0, 1}* mean an arbitrary sequence of 0's and 1's?
2. What is a DFA? Presumably this acronym is unpacked in your textbook, but we don't have access to that information.
3. What does the notation $\hat \delta (p, w)$ mean?