How does "δ:Q×Σ→Q" read in the definition of a DFA (deterministic finite automaton)?

Question

How do you say δ: Q × Σ → Q in English? Describing what × and → mean would also help.

Answer 1

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:

+----+----+
|Q\Σ | 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.

Answer 2

Sorry if the terms are not exactly right (in English). I've studies automaton theory about 3 or 4 years ago in a different language so the terms might not be exactly accurate.

δ is like a partial function with two arguments which takes as an input a state (that automaton's state; element of Q) and an "input action" (element of Σ which is the alphabet accepted by the automaton*) and produces the new state the automaton should have after providing it the input action.

Basically this can be read as:

delta transition function defined on the Q set of automaton states and sigma alphabet

The × in the formula represents a Cartesian product of the states and actions sets, while the → represents that what is returned by the function belongs to the set following it (in your case, Q).

*not to be confused with the language accepted by the automaton which would be all the "input actions" sequences that have valid transitions (i.e. δ(stateX, input) is defined) and lead the automaton into a final "acceptable" state.