δ is like a mathematical function called the transition function . Something like.
z = f(x, y)
A function in mathematical defines mapping of elements in one set to another set. In function set of input arguments are called Domain of a function and output is the rage.
[ANSWER] ??
In expression "δ:Q×Σ → Q",
× means Cartesian product (that is a set), and → is a mapping.
"δ:Q×Σ → Q" says δ is a transition function that defined mapping from Q×Σ to Q.
Where, Domain of δ is Q × Σ and Range is Q.
Note: Cartesian Product itself a mathematical that all possible order pair (mapping) between two sets.
You can also say:
δ is a transition function that defined mapping between(or say associates) Cartesian product of set of statesQ and language symbolsΣ into set of stateQ. This is abbreviated by δ: Q×Σ → Q
Here, Q is finite set of states and Σ is a finite set of language symbols.
Additionally in any automated you can represent transition function in tree ways.
1. Transition Table
2. Transition graph or say state diagram.
3. Transition function: a finite set of mapping rules. e.g. {δ(q0, a) → q1, δ(q1, a) → q2}
All for same purpose define maping
In DFA. δ:Q×Σ → Q can also be written like δ(Q,Σ) → Q It's similar to function. In δ function two input arguments are state Q and a language symbol Σ and returned value is Q.
What is meaning of δ(Q,Σ) → Q
Suppose in your set of transition function δ you have an element δ(q0, a) → q1 this means.
If the present state is q0 then by consuming a symbol you can shift to state q1. And the state-diagram for δ(q0, a) → q1:
(q0)---a---?(q1)
and transition table is:
+----+----+
|QΣ | a |
+----+----+
| q0 | q1 |
+----+----+
and all defines mapping (q0, a) to (q1).
Some authors write δ ? Q×Σ → Q in formal DFA definition that means δ is a Partial function (not defined on full Domain Q×Σ). We can always defined δ on the full domain that is required sometime for example to find complement DFA.
Here(Complement DFA), I wrote two DFAs for the same language one is partial DFA other is complement DFA.
