Page 85 - Programmable Logic Controllers, Fifth Edition

A AB A A + B
AND Y = AB + C OR Y = (A + B) C
B OR B AND
C C

A AB A A + B
NAND Y = AB + C NOR Y = (A + B) C
B OR B AND
C C
Figure 4-17 Logic operators used in combination to form Boolean equations.

B
ASSOCIATIVE LAW
C
(A + B) + C = A + (B + C)
(A ⋅ B) ⋅ C = A ⋅ (B ⋅ C)
D

DISTRIBUTIVE LAW Y
A Output
A ⋅ (B + C) = (A ⋅ B) + (A ⋅ C ) Inputs
A + (B ⋅ C) = (A + B) ⋅ (A + C ) Figure 4-19 Logic gate circuit developed from the Boolean
This law holds true only in expression Y = A(BC + D).
Boolean algebra.

1 - OR gate with input C and output from previous
AND gate

4.4 Developing Logic Gate Circuits Figure 4-19 shows a logic gate circuit developed from
from Boolean Expressions the Boolean expression Y = A(BC + D). The procedure
is as follows:
As logic gate circuits become more complex, the need to Boolean expression: Y = A(BC + D)
express these circuits in Boolean form becomes greater. Gates required: (by inspection)
Figure 4-18 shows a logic gate circuit developed from the 1 - AND gate with inputs B and C
Boolean expression Y = AB + C. The procedure is as 1 - OR gate with inputs BC and D
follows: 1 - AND gate with input A and the output from the

Boolean expression: Y = AB + C OR gate
Gates required: (by inspection)
1 - AND gate with input A and B 4.5 Producing the Boolean Equation
for a Given Logic Gate Circuit

A simple logic gate is quite straightforward in its opera-
tion. However, by grouping these gates into combinations,
A it becomes more difficult to determine which combinations
of inputs will produce an output. The Boolean equation for
B the logic circuit of Figure 4-20 is determined as follows:

• The output of the OR gate is A + B
Y –
C Output • The output of the inverter is D
Inputs
• Based on the input combination applied to the AND
Figure 4-18 Logic gate circuit developed from the Boolean gate the Boolean equation for the circuit is Y = C
–
expression Y = AB + C. D (A +B)

66 Chapter 4 Fundamentals of Logic

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