How do quadruples or quintuples describe the instructions of a Turing machine?

In automata theory, a Turing machine is defined by a finite set of instructions that dictate its operations based on its current internal state and the symbol scanned on its tape. These instructions are formally represented as either quadruples or quintuples of symbols.1

Defining Instructions

To formalize the machine's operation, a finite list of symbols is used to represent:
* Internal states: Denoted as $q_1,q_2,\dots ,q_n$.1
* Alphabet: A finite set of symbols (e.g., $a,b,c,\dots ,\lambda$) that can be printed on the tape.1
* Tape movement: Symbols such as $R$ (right) and $L$ (left) designate the direction of the tape head.1

Quadruples vs. Quintuples

Turing originally defined these instructions using quintuples, which provide a complete description of the machine's possible steps. However, it is mathematically equivalent to define a Turing machine using quadruples.1

Because the list of these instructions is finite, all possible Turing machines are at most countable in number. This finiteness allows for the existence of a "universal" machine capable of simulating any other Turing machine by encoding its specific set of quadruples onto its own tape.1