Boolean algebraUsing Boolean representations for logic operations, some of the mathematical proper-ties of multiplication, addition, and negation can be applied to form Boolean equations.The logical combinations on either side of any equation are equivalent.In some ways, Boolean algebra differs from conventional algebra. You must uselogic rules rather than “regular” rules for addition, additive inverse (negation), andmultiplication. Using these rules, certain facts, called theorems, can be derived.Boolean theorems all take the form of equations. Some common Boolean theorems arelisted in Table 30-7.564 Basic digital principlesTable 30-7 Common theorems in Boolean algebra.X0XOR identityX1XAND identityX11X00XXXXXX(X)XDouble negationX(X)XX(X)0XYYXCommutativity of ORXYYXCommutativity of ANDXXYXX(Y)YXYXYZ(X + Y) ZX(YZ) Associativity of ORXYZ(XY)ZX(YZ)Associativity of ANDX(YZ)XYXZDistributivity(XY)(X)(Y)DeMorgan’s Theorem(XY)X YDeMorgan’s TheoremBoolean algebra is less messy than truth tables for designing and evaluating logiccircuits. Some engineers prefer truth tables because the various logic operations areeasier to envision, and all the values are shown for all logic states in all parts of a digitalcircuit. Other engineers would rather not deal with all those ls and 0s, nor cover wholetabletops with gigantic printouts. Boolean algebra gets around that.For extremely complex logical circuits, computers are used as an aid in design.They’re good at combinatorial derivations and optimization problems that would be un-economical (besides tedious) if done by a salaried engineer.The flip-flopSo far, all the logic gates discussed have outputs that depend only on the inputs. Theyare sometimes called combinational logic gates, because the output state is simply afunction of the combination of input states.