Boolean Algebra and Logic Circuits — FAQ

For expressions with 2 or 3 variables, algebra is often just as fast. However, for 4 variables, it becomes very difficult to be certain you have found the absolute simplest form using algebra alone. A K-map provides a systematic, visual method that guarantees the most simplified sum-of-products expression if you follow the rules correctly.

A half adder can only add two single bits, like the two bits in the least significant column (the 'ones' column) of a binary addition. A full adder can add three bits: two input bits plus a carry-in from the previous column. This makes it possible to chain full adders together to create a multi-bit adder (e.g., an 8-bit adder is made of one half adder and seven full adders).

In an SR flip-flop made from NOR gates, setting S=1 and R=1 forces both the output Q and its complement $\overline{Q}$ to be 0. This violates the fundamental rule that Q and $\overline{Q}$ must be opposites. The state is also unstable; if S and R return to 0 simultaneously, the final state of the flip-flop is unpredictable (a 'race condition').

The clock signal provides synchronisation. The D-type flip-flop only pays attention to its data (D) input at a very specific moment—the instant the clock signal transitions from low to high (a rising edge) or high to low (a falling edge). At all other times, the D input can change without affecting the stored output Q. This allows many flip-flops in a larger circuit (like a processor register) to change state simultaneously and in an orderly fashion.