这两个离散有限自动机有何不同？

Question

I have the discrete finite automaton defined by the following statement:

{ω | ω is any string not in a* ∪ b*}

For some reason I'm just not understanding the "a* ∪ b*" part. I know what a union is, but how is this different from a* b*? Is the resulting DFA of these two statements the same? I need to first create the DFA for the complement of this language, and then use that DFA to create the DFA of the above language based on that.

Can someone help me understand this?

Answer 1

Let's break down the definitions and then it'll be simpler to see the difference between a* ∪ b* and a* b* .

• The set x* refers to any non-negative number (including zero) of times of the symbol x . This includes the empty string ( ε ), x , xx , xxx and so forth.
• The set XY refers to any symbol that is in X followed by Y
  • If X = {1, 2} and Y = {a, b} , then XY = {1a, 1b, 2a, 2b}
• The set X ∪ Y refers to the union of the sets described by X and Y . This includes any symbols that may be in set X OR Y but not necessarily both.
  • If X = {1, 2} and Y = {a, b} , then X ∪ Y = {1, 2, a, b}

From the above we can tell deduce that the set of elements in a* b* is any element in the set a* followed by any element in the set b* (remember since we're using the * notation, the empty string is included). a* = {ε, a, aa, aaa, aaaa, ... } and b* = {ε, b, bb, bbb, bbbb, ... } . Therefore a* b* = {ε, a, b, ab, aa, aab, aabb, bb, abb, aabb, ...} .

Similarly we now know that a* ∪ b* includes any elements in the set a* OR b* . Therefore a* ∪ b* = {ε, a, b, aa, bb, aaa, bbb, aaaa, bbbb, ...} . Notice there isn't any element that has both the symbols a AND b because that is not in the set.

Finally, you can ask what are the elements that are in a* b* BUT are not in a* ∪ b* . a* b* \\ a* ∪ b* = { x ∈ a* b* | x ∉ a* ∪ b*} = { ab, abb, abbb, ... aab, aabb, aabbb, ..., aaab, aaabb, ... } a* b* \\ a* ∪ b* = { x ∈ a* b* | x ∉ a* ∪ b*} = { ab, abb, abbb, ... aab, aabb, aabbb, ..., aaab, aaabb, ... } . These are the elements with both symbols a and b in them.

Answer 2

a* U b* is the language of all strings of a only along with all strings of b only: {empty, a, b, aa, bb, aaa, bbb, ...} . The strings not in this language are those strings that contain not only a s or b s but both: {ab, ba, aab, aba, baa, abb, bab, bba, ...} . a*b* is yet another language which consists of all the strings where any a s come before any b s: {empty, a, b, aa, ab, bb, ...} . a*b* is a superset of a* U b* but it is not equal to it. It is neither a subset nor a superset of the language of strings not in a* U b* but it does overlap in many places. Since all three languages are distinct, all three have distinct DFAs.