2.1 BASIC OF LOGIC GATES.
The variables and function have only one value, 0 and 1. The complement of a variable is shown by a bar over the letter such as ( ) or apostrophe (A’).
Algebraic Expressions :
1) AND - (F = A.B) OR (F = AB)
2) OR - (F = A+B)
3) NOT - (F = A’)
4) NAND - (F = (A.B)’)
5) NOR - (F = (A+B)’)
6) XOR - (F = A.B’ + A’.B)
2.2 COMBINATIONAL CIRCUITS.
v A logic block contains no memory and computers the output given the current inputs.
v Can be defined in three ways :
1. Truth Table – The truth table shows many possible combinations of input values, in tabular from between the input values and the result of a specific Boolean operator or combinations on the input variables.
2. Graphical Symbols – The layout of connected gates that represent the logic circuit.
3. Boolean Equations – Boolean functions that consist possible combination of inputs that produce an output signal.
2.2.1 BOOLEAN EQUATION FORMS.
A Boolean algebra is the combinations of variables and operators. Typically, it has one or more inputs and produces an output in the range of 0 or 1. The complement of a variable is shown by a bar over the letter.
All Boolean equation can be represented in two forms :
1) Sum-Of-Products (SOP)
Combination of input values that produce 1s is convert into equivalent variables, ANDed together then ORed with other combination variables with the same output.
2) Product-Of-Sums (POS)
Input combinations that produce 0 in sum terms (ORed variables) are ANDed together.
2.2.2 SIMPLIFICATION OF BOOLEAN EQUATION.
There are two ways to simplify Boolean equation, Laws of Boolean Algebra and
1) Laws Of Boolean Algebra – Rules to simplify Boolean expression.
2) Karnaugh Map – A grid-like representation of a truth table.
Example 1 :
F = A’B’ + ABC’D
A
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B
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C
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D
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F
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0
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0
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0
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0
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1
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0
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0
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0
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1
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1
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1
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0
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1
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0
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1
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1
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1
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0
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1
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0
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0
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0
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1
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0
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1
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0
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0
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1
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1
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0
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0
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0
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1
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1
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1
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0
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1
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0
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0
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0
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0
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1
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0
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0
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1
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0
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1
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0
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1
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0
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1
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1
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0
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1
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1
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0
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1
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1
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1
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1
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0
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1
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1
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1
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0
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SOP expression :
F = (A’B’C’D’) + (A’B’C’D) + (A’B’CD’) + (A’B’CD) + (ABC’D)
*Note that the method is reversible. You can find the SOP expression from the truth table or build the truth table from the expression.
Example 2 :
F = (A+B+C) (A+B+C’) (A+B’+C) (A’+B+C)
A
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B
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C
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F
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0
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0
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0
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0
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0
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0
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1
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0
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0
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1
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0
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0
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0
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1
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1
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1
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1
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1
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1
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0
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1
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1
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0
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1
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1
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1
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1
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1
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POS expression :
F = (A+B+C) (A+B+C’) (A+B’+C) (A’+B+C)
Note : This is not simplified version
*Note that the method is reversible. You can find the POS expression from the truth table or build the truth table from the expression.
2.2.2.1 LAWS OF BOOLEAN ALGEBRA.
Basic Laws of Boolean Algebra :
1) A . 0 = 0
2) A .1 = A
3) A + 0 = A
4) A + 1 = 1
5) A + A = A
6) A + A’ = 1
7) A . A = A
8) A . A’ = 0
9) A’’ = A
10) A + AB = A
11) A + A’B = A + B
12) (A + B) (A + C) = A + BC
2.2.2.1.1 DE MORGAN’S LAW.
There is a trick to remember the laws. An aid to memory is :
*Note : If you break the line, you change the sign.
(A.B)’ = A’ + B’
(A+B)’ = A’. B’
2.2.2.2 KARNAUGH MAP.
The Karnaugh map or K-Map provides a simple and straight-forward method of minimizing Boolean expressions. The only limitation is that it will be ineffective for more than four (4) inputs.
Note :
For n inputs, there are 2^ cells needed to build the K-Map
Example :
F = A + B
· The Truth Table :
A
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B
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A+B
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A’B’
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0
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0
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0
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A’B
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0
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1
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1
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AB’
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1
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0
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1
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AB
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1
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1
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1
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· Karnaugh Map :
B\A
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A’
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A
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B’
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0
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1
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B
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1
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1
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· Boolean Law :
F = A’B + AB’ + AB
= A’B + AB + AB’ + AB
= B ( A’ + A ) + A ( B’ + B )
= B + A
= A + B
2.3 UNIVERSAL GATES.
Gates that can be used to implement any gates like AND, OR and Not or any combination of these basic gates are called universal gates. NAND and NOR are such examples.
2.3.1 NAND GATE.
A NAND gate ( Negated AND or NOT AND ) is a logic gate which produces an output that is false only if all its inputs are true.
2.3.2 NOR GATE.
A NOR gate ( Negated OR or NOT OR ) is a logic gate which produces a HIGH output (1) results if both the inputs to the gate are LOW (0); a LOW output (0) results if one or both input is HIGH (1).
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