Boolean Algebra


   
 
Application to Switching Circuits (Contd...)
Logic Gates
 
The basic building blocks for a complex digital system are known as logic gates.
 
A gate is a simple electronic circuit or device that performs logical functions. It has one or more inputs and one output gate and are also called binary logic gates, as 1 and 0 are the inputs and outputs.
 
We shall discuss the three basic gates known as 'AND', 'OR' and 'NOT' gates whose input - output tables are similar to conjunction, disjunction and negation respectively with T = 1 and F = 0.
 
AND Gate
 
An 'AND' gate is a Boolean function with two input variables x1, x2, defined by
 
f(x1, x2) = x1, x2
 
where x1, x2 {0, 1}
 
The input-output table for 'AND' gate is given below, which is similar to Table 3 (A circuit with two switches connected in series)
 
Table 10
 
 
The following figure represents 'AND' gate
 
 
OR Gate
 
An 'OR' gate is a Boolean function with two variables defined by
 
f(x1, x2) = x1 + x2 x1, x2 {0, 1}
 
 
The above figure shows the 'OR' gate and the input-output table for OR gate is given in Table 11 which is similar to Table 6 (A circuit with two switches connected in parallel)
 
Table 11
 
 
NOT Gate
 
A NOT gate is the Boolean function defined by
 
f(x) = x' x{0, 1}
 
 
NOT gate is shown in figure and the input/output table is given in Table 12.
 
Table 12
 
 
Example:
 
Find the output for the given input.
 
 
x1 = 1, x2 = 1 from the circuit given in figure.
 
The Boolean function for the given circuit is
 
S = f(x1, x2) = (x1 . x2)' + x1
 
If x1 = 1, x2 = 1
 
S = (x1 . x2)' + x1 = (1 . 1)' + 1
 
= (1)' + 1 = 0 + 1
 
= 1
 
Combinatorial Circuit
 
A circuit in which the output is uniquely defined for each combination of inputs x1, x2, x3 is called a combinatorial circuit.
 
Equivalent Circuit
 
Two combinatorial circuits, each having inputs x1, x2, x3, x4,……xn and a single output are said to be equivalent if their input/output tables are identical.
 
Example:
 
Simplify the combinatorial circuit for the given figure.
 
 
The given circuit is x1.x2 + x1'x2 + x1x2'
 
= x2(x1 + x1') + x1x2' Axiom 4(a)
 
= x2 . 1 + x1x2' Axiom 5(a)
 
= x2 + x1x2' Axiom 2(b)
 
= x1x2' + x2 Axiom 3(a)
 
The simplified combinatorial circuit is
 
 
 
     
   
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