# Boolean ```Boolean Theorems and Boolean Algebra
We have seen how Boolean expressions can be used to express a logic operation mathematically.
We can also use these expressions combined with Boolean theorems to simplify a logic circuit.
The Boolean theorems are grouped into two categories below. Table 1 lists the single-variable
theorems. Table 2 lists the theorems relating expressions with multiple variables. Table 3 lists
two special multivariable theorems which express how to change an AND expression to an OR
expression and vice versa. DeMorgan’s theorems in Table 3 are also covered in your textbook.
Note that AB and A  B are equivalent representations for the AND function. The dot notation
is used below to avoid confusion when combining numbers and variables, ie, to differentiate the
expression x 0 from the variable x0.
1. x  0  0
5. x  0  x
x
0
x
0
2. x 1  x
x
0
6. x 1  1
x
x
x
1
1
3. x  x  x
4. x  x  0
1
x
7. x  x  x
x
x
0
8. x  x  1
x
x
x
1
Table 1: Single-variable Theorems
9. x  y  y  x
13a. x( y  z )  xy  xz
10. x  y  y  x
13b. ( w  x)( y  z )  wy  xy  wz  xz
11. x  ( y  z )  ( x  y )  z  x  y  z
14. x  xy  x
12. x( yz )  ( xy ) z  xyz
15a. x  xy  x  y
15b. x  xy  x  y
Table 2: Multivariable Theorems
16. ( x  y )  x  y
EE302 – Spring 2004
Boolean Theorems and Boolean Algebra
17. ( x  y )  x  y
Table 3: DeMorgan’s Theorems
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These theorems can be used to simplify expressions as shown in the examples below. In all of
the following examples, we refer to the theorems above by number for simplicity.
Example 1
Simplify the expression z  ABD  ABD to a sum of products (SOP) form.
z  ABD  ABD
by theorem 13
z  AB ( D  D)
by theorem 8
z  AB (1)
 z  AB
by theorem 2
z  AB
Example 2
Simplify the expression z  ( A  B)( A  B) to a sum of products form.
z  ( A  B)( A  B)
z  AA  AB  BA  BB
z  0  AB  BA  B
z  B( A  A  1)
z  B(1)
zB
by theorem 13
by theorems 3 and 4
by theorem 13
by theorem 6
by theorem 2
zB
Example 3
Simplify the expression z  ( A  C)  ( B  D) to a sum of products form.
z  ( A  C)  ( B  D)
z  ( A  C )  ( B  D) by theorem 17
z  ( AC )  ( BD)
z  ( AC )  ( BD)
by theorem 16
 z  AC  BD
Example 4
Determine the output expression of the circuit in Figure 1 and simplify to a sum of products.
B
A
C
Figure 1: Logic Circuit
z  A B C
z
z  A B C
z  A B C
by analysis
by theorem 17
 z  A B C
Note that in Example 4 we have changed an AND expression (since a NAND gate is an AND
gate with an inverter after it) to an OR expression. This conversion leads directly to our ability
to implement any logic expression using any universal gate, eg, NAND or NOR. Universal gates
are discussed further in your textbook.
EE302 – Spring 2004
Boolean Theorems and Boolean Algebra
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