2 Logic Gates and Boolean Algebra Boolean algebra is an important tool in describing, analyzing, designing, and implementing digital circuits.
3 Boolean Constants and Variables Boolean algebra allows only two values; 0 and 1.Logic 0 can be: false, off, low, no, open switch.Logic 1 can be: true, on, high, yes, closed switch.Three basic logic operations: OR, AND, and NOT.
4 Truth TablesA truth table describes the relationship between the input and output of a logic circuit.The number of entries corresponds to the number of inputs. For example a 2 input table would have 22 = 4 entries. A 3 input table would have 23 = 8 entries.
5 Truth Tables Examples of truth tables with 2, 3, and 4 inputs.
6 Boolean algebra Used to design and analysis digital circuit Invented by George BooleUses variables and operationsVariable may take value TRUE or FALSELogical operations are AND,OR ,and NOT
7 Boolean Laws A. 0 = 0 A + 0 =A A . 1 =A A + 1 = 1 Ã + 1 = 1 A . A = A -
8 Boolean Laws A . B = B.A A + B = B +A A.(B.C)=(A.B).C A +(B+C)=(A+B)+C A.(B+C)=A.B + A.CA+(B.C)=(A+B).(A+C)A.(A+B)=A
9 Boolean Laws A + AB =A A.(Ã + B) = AB A + ÃB= A + B Ã + AB = Ã + B
10 Duality Swap 0 & 1, AND & OR Principle of Duality (Metatheorem) Result: Theorems still truePrinciple of Duality (Metatheorem)Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and • and + are swapped throughout.Fully parenthesized before taking its duality
11 Duality Theorem Find dual of A + AB = A Step 1 = A .(A . B) = A A. (A + B) = A is dual of A + AB =A
12 DeMorgan’s Theorem Equation: Equation: The complement of two or more ANDed variables is equivalent to the OR of the complements of the individual variablesEquation:The complement of two or more ORed variables is equivalent to the AND of the complements of the individual variablesEquation:
13 A + AB = A(1 +B) DISTRIBUTIVE LAW = A∙1 (1+B)=1 = A A∙1 = A Boolean Rules A + AB = AProof:A + AB = A(1 +B) DISTRIBUTIVE LAW= A∙ (1+B)=1= A A∙1 = A
14 (A + B)(A + C) = A + BC PROOF (A + B)(A +C) = AA + AC +AB +BC DISTRIBUTIVE LAW= A + AC + AB + BC= A(1 + C) +AB + BC= A.1 + AB + BC= A(1 + B) + BC= A.1 + BC= A + BC
15 OR Operation With OR Gates The Boolean expression for the OR operation isX = A + BThis is read as “x equals A or B.”X = 1 when A = 1 or B = 1.Truth table and circuit symbol for a two input OR gate:
16 OR Operation With OR Gates The OR operation is similar to addition but when A = 1 and B = 1, the OR operation produces = 1.In the Boolean expressionx=1+1+1=1We could say in English that x is true (1) when A is true (1) OR B is true (1) OR C is true (1).
17 AND Operations with AND gates The Boolean expression for the AND operation isX = A • BThis is read as “x equals A and B.”x = 1 when A = 1 and B = 1.Truth table and circuit symbol for a two input AND gate are shown. Notice the difference between OR and AND gates.
18 Operation With AND Gates The AND operation is similar to multiplication.In the Boolean expressionX = A • B • CX = 1 only when A = 1, B = 1, and C = 1.
19 NOT Operation The Boolean expression for the NOT operation is This is read as:x equals NOT A, orx equals the inverse of A, orx equals the complement of A
20 NOT OperationTruth table, symbol, and sample waveform for the NOT circuit.
21 Describing Logic Circuits Algebraically The three basic Boolean operations (OR, AND, NOT) can describe any logic circuit.If an expression contains both AND and OR gates the AND operation will be performed first, unless there is a parenthesis in the expression.
22 Describing Logic Circuits Algebraically Examples of Boolean expressions for logic circuits:
23 Describing Logic Circuits Algebraically The output of an inverter is equivalent to the input with a bar over it. Input A through an inverter equals A.Examples using inverters.
24 NOR Gates and NAND Gates Combine basic AND, OR, and NOT operations.The NOR gate is an inverted OR gate. An inversion “bubble” is placed at the output of the OR gate.The Boolean expression is,
25 NOR Gates and NAND Gates The NAND gate is an inverted AND gate. An inversion “bubble” is placed at the output of the AND gate.The Boolean expression is
26 NAND and NOR GatesNAND and NOR gates can greatly simplify circuit diagrams. Use these gates wherever you could use AND, OR, and NOT.ABAB1NANDABAB1NOR
27 Universality of NAND and NOR Gates NAND or NOR gates can be used to create the three basic logic expressions (OR, AND, and INVERT)This characteristic provides flexibility and is very useful in logic circuit design.