the transition function
/ delta = the symbol δ /
The transition function, written δ (delta), is the rulebook of the machine: it says what to do on every single move. Its job is to answer one kind of question — 'I am in this state and I just read this symbol; where do I go next?' — and to answer it the same way every time. Picture a flat table you consult: look up the row for your current state, the column for the symbol you just read, and the cell tells you the one state to move to.
Formally δ is a function from pairs (state, symbol) to states: δ : Q × Σ -> Q, and we write δ(q, a) = p. The two facts hidden in that signature are exactly what make a DFA deterministic and total. Deterministic: the output is a single state p, never a set and never a choice. Total: δ is defined for every state and every symbol — there is no missing entry, no situation where the machine gets stuck not knowing what to do. Every cell of the table is filled.
Everything the machine ever does is just δ applied over and over. To process a string you start in q0 and apply δ once per symbol, threading the output state of one step into the input state of the next. Designing a DFA really means designing its δ: deciding, for each state and symbol, which next state correctly continues the bookkeeping you intend.
Parity of 1-s over {0,1}: states Even, Odd. δ(Even,0)=Even, δ(Even,1)=Odd, δ(Odd,0)=Odd, δ(Odd,1)=Even. Reading a 0 never flips parity; reading a 1 always flips it. Four entries cover both states times both symbols — total and deterministic.
δ(q, a) = p: from state q reading symbol a, move to the one state p.
For a DFA δ outputs one state. The NFA version sends (state, symbol) to a set of states (possibly empty), which is the real difference between the two models — but that lives in the NFA topic, not here.