Who's lying?

As the only case with at least 3 $\color{#20A900}True$ statements, $\boxed{Betty}$ broke the vase.

Since D and E say the same thing, they are either both liars or not. Since them both lying would imply that A is also lying, which would make 3 liars when there can only be 3, they are truthful. Therefore, E and D are true, making A true. B and C are liars, and B broke the vase.

Five statements are:

Alice: Betty broke it!

Betty: Alice is not telling the truth.

Cathy: I did it!

David: I did not break it and Betty is lying!

Edward: David is telling the truth!

If Alice broke the vase,

Alice's statement is $\color{#D61F06}false$ , Betty's statement is $\color{#20A900}true$ , Cathy's statement is $\color{#D61F06}false,$ David's statement is $\color{#D61F06}false$ , Edward's statement is $\color{#D61F06}false.$ So it is not Alice.

If Betty broke the vase,

Alice's statement is $\color{#20A900}true,$ Betty's statement is $\color{#D61F06}false,$ Cathy's statement is $\color{#D61F06}false$ , David's statement is $\color{#20A900}true$ , Edward's statement is $\color{#20A900}true.$ So it is Betty.

If Cathy broke the vase,

Alice's statement is $\color{#D61F06}false,$ Betty's statement is $\color{#20A900}true$ , Cathy's statement is $\color{#20A900}true$ , David's statement is $\color{#D61F06}false$ , Edward's statement is $\color{#D61F06}false$ . So it is not Cathy.

If David broke the vase,

Alice's statement is $\color{#D61F06}false$ , Betty's statement is $\color{#20A900}true$ , Cathy's statement is $\color{#D61F06}false,$ David's statement is $\color{#D61F06}false$ , Edward's statement is $\color{#D61F06}false.$ So it is not David.

If Edward broke the vase,

Alice's statement is $\color{#D61F06}false$ , Betty's statement is $\color{#20A900}true$ , Cathy's statement is $\color{#D61F06}false,$ David's statement is $\color{#D61F06}false$ , Edward's statement is $\color{#D61F06}false.$ So it is not Edward.

Since at least three children are telling the truth, $\boxed{\text{Betty}}$ broke the vase.

Note that D and E are either both true or both false.

Note also that of A and B , one must be true and the other must be false.

Since at most two of the statements may be false, we conclude that D and E are actually true.

Therefore B is lying, so A is telling the truth: $\text{Betty}$ broke the vase.

Let each letter A, B, C, D and E be the truth value of each respective child's statement.

You can construct 4 compound statements: (~ is negation, <=> is material equivalence) A <=> ~B C <=> ~A D <=> ~B E <=> D

Finally, you can derive that: A <=> D <=> E

Since you've formed at least 3 statements that are consistent with each other, you can assume them true. Hence, from A, Betty broke the vase.

No need of tables.. Just simplify their statements:

A says B

B says !B

C says C

D says !D && !(!B) --> !D && B --> B

E says B

3 children stated that B broke it..

I wrote out each child's proposition, along with other propositions it implied. I treated it as "what does each letter say the solution is?"

A- B (A says B is the solution) B- A is false C- C D- ~D, B is false, (implies B) E- D is true, (implies ~D, B is false.)

B appears in 3 propositions, so I started there. D and E don't conflict with each other, so they might both be true. Following this assumption, B is the solution and, because B is false, A is true. A says B is the solution. Here we have 3 true statements (A, D, and E) that point to B as the solution. It does not matter if C is true or not because the question describes "at least three truth tellers."

If at least three statements are true, then there can be no more than two false statements. One of Alice/Betty must be false, since they contradict one another. The other three must either all be true or have exactly one false statement between them. In order for Edward's statement to be true, David's must also be true, making Cathy's statement false. This makes Betty the vase-breaker.