CHAPTER 7. DIGITAL CIRCUITS1347.2Boolean AlgebraThe binary 0 and 1 states are naturally related to the true and false logic variables. We willﬁnd the following Boolean algebra useful. Consider two logic variables A and B and theresult of some Boolean logic operation Q. We can deﬁneQ≡ A AND B≡ A· B.(7.15)Q is true if and only if A is true AND B is true.Q≡ A OR B≡ A + B.(7.16)Q is true if A is true OR B is true.Q≡ NOT A≡ A.(7.17)Q is true if A is false.A useful way of displaying the results of a Boolean operation is with a truth table. Wewill make extensive use of truth tables later. If no “–” is available on your text processor orcircuit drawing program an “N ” can be used, ie. A≡ NA .We list a few trivial Boolean rules in table 7.2.Table 7.2: Properties of Boolean Operations.A· 0=0A +0=AA· 1=AA +1=1A· A=AA + A=AA· A=0A + A=1The Boolean operations obey the usual commutative, distributive and associative rulesof normal algebra (table 7.3).We will also make extensive use of De Morgan’s theorems (table 7.4).7.3Logic GatesElectronic circuits which combine digital signals according to the Boolean algebra are referredto as logic gates; gates because they control the ﬂow of information. Positive logic is anelectronic representation in which the true state is at a higher voltage, while negative logichas the true state at a lower voltage. We will use the positive logic type in this course.In digital circuits all inputs must be connected.