How did they get the equations break down simply?

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Latches and Flip Flop

Just like logic gates, the latches and flip-flops are the important elements of an electronic circuit. The logic gates form the important part of combinatorial circuits, while the latches and flip-flops form the essential part of a sequential circuit. Sequential circuits form the important concept of automata theory in electronics and electrical engineering, these circuits are a kind of logic circuit whose output depends on both present input and past inputs, that is, the input history.