Page 54 - DCAP108_DIGITAL_CIRCUITS_AND_LOGIC_DESIGNS

Unit 3: Boolean Algebra

Notes
Figure.3.8 (a): Equivalent Circuits Implied by Theorem (1):
(b) Alternative Symbol for the NOR Function

x
x x
x+y y x . y=x+y
y y
a
()

x
x . y=x+y
y
()
b
Figure 3.9 (a): Equivalent Circuits Implied by Theorem (2);
(b) Alternative Symbol for the NAND Function
x
x x
xy y x . y=xy
y y
()
a

x
x . y=xy
y

()
b
What this means is that an AND gate with INVERTERs on each of its inputs is equivalent to a
NOR gate. In fact, both representations are used to represent the NOR function. When the AND
gate with inverted inputs is used to represent the NOR function, it is usually drawn as shown in
Figure 3.8 (b), where the small circles on the inputs represent the inversion operation.

Now consider theorem (2),
x y = x + y
The left side of the equation can be implemented by a NAND gate with inputs x and y. The
right side can be implemented by first inverting inputs x and y and then putting them through
an OR gate. These two equivalent representations are show in Figure 3.9(a). The OR gate with
INVERTERS on each of its inputs is equivalent to the NAND gate. In fact, both representations
are used to represent the NAND function, while the OR gate with inverted inputs is used to
represent the NAND function.

Simplify the expression z = (A + C ) (B + D ) to one having only single variables
•
inverted.
Solution:
Using theorem (2), we can rewrite this as

z = (A + C ) + (B + D )
We can think of this as breaking the large inverter sign down the middle and changing
the AND sign (•) to an OR sign (+). Now the term (A + C ) can be simplified by applying
theorem (1). Likewise, (B + D ) can be simplified:

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