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|Combinational Logic:||[Basic Gates] [Derived Gates] [The XOR Function] [Binary Addition] [Negative Numbers and Binary Subtraction] [Multiplexer] [Decoder/Demultiplexer] [Boolean Algebra]|
|Sequential Logic:||[RS NAND Latch] [RS NOR Latch] [Clocked RS Latch] [RS Flip-Flop] [JK Flip-Flop] [D Latch] [D Flip-Flop] [Flip-Flop Symbols] [Converting Flip-Flop Inputs]|
|Alternate Flip-Flop Circuits:||[D Flip-Flop Using NOR Latches] [CMOS Flip-Flop Construction]|
|Counters:||[Basic 4-Bit Counter] [Synchronous Binary Counter] [Synchronous Decimal Counter] [Frequency Dividers] [Counting in Reverse] [The Johnson Counter]|
|Registers:||[Shift Register (S to P)] [Shift Register (P to S)]|
|The 555 Timer:||[555 Internals and Basic Operation] [555 Application: Pulse Sequencer]|
One of the primary requirements when dealing with digital circuits is to find ways to make them as simple as possible. This constantly requires that complex logical expressions be reduced to simpler expressions that nevertheless produce the same results under all possible conditions. The simpler expression can then be implemented with a smaller, simpler circuit, which in turn saves the price of the unnecessary gates, reduces the number of gates needed, and reduces the power and the amount of space required by those gates.
One tool to reduce logical expressions is the mathematics of logical expressions, introduced by George Boole in 1854 and known today as Boolean Algebra. The rules of Boolean Algebra are simple and straight-forward, and can be applied to any logical expression. The resulting reduced expression can then be readily tested with a Truth Table, to verify that the reduction was valid.
The rules of Boolean Algebra are:.
0·0 = 0 A·0 = 0 1·0 = 0 A·1 = A 0·1 = 0 A·A = A 1·1 = 1 A·A' = 0
0+0 = 0 A+0 = A 1+0 = 1 A+1 = 1 0+1 = 1 A+A = A 1+1 = 1 A+A' = 1
0' = 1 A'' = A 1' = 0
(A·B)·C = A·(B·C) = A·B·C (A+B)+C = A+(B+C) = A+B+C
A·(B+C) = (A·B) + (A·C) A+(B·C) = (A+B) · (A+C)
A·B = B·A A+B = B+A
AB = A·B A·B+C = (A·B) + C A+B·C = A + (B·C)
(A·B)' = A' + B' (NAND) (A+B)' = A' · B' (NOR)
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