Another way to find a Boolean expression that represents a Boolean function is to form a Boolean product of Boolean sums of literals. Exercises 1–5 are concerned with representations of this kind. 1....

Another way to find a Boolean expression that represents a Boolean function is to form a Boolean product of Boolean sums of literals. Exercises 1–5 are concerned with representations of this kind.

1. Find a Boolean sum containing either
x
or
 either
y
or  and either
z
or

that has the value 0 if and only if

a)
x
=
y
= 1, z
= 0.

b)
x
=
y
=
z
= 0.

c)
x
=
z
= 0, y
= 1.

2. Find a Boolean product of Boolean sums of literals that has the value 0 if and only if
x
=
y
= 1 and
z
= 0,
x
=
z
= 0 and
y
= 1, or
x
=
y
=
z
= 0.

3. Show that the Boolean sum
y
₁+
y
₂+· · ·+y_n
, where
y_i
=
x_i
or
y_i
=
x_i
, has the value 0 for exactly one combination of the values of the variables, namely, when
x_i
= 0 if
y_i
=
x_i
and
x_i
= 1 if
y_i
=
x_i
. This Boolean sum is called a maxterm.

4. Show that a Boolean function can be represented as a Boolean product of maxterms. This representation is called the product-of-sums expansion or conjunctive normal form of the function.

5. Find the product-of-sums expansion of each of the Boolean functions in Exercise 6.

Exercise 6 Find the sum-of-products expansions of these Boolean functions.

a)
F(x, y, z)
=
x
+
y
+
z

b)
F(x, y, z)
=
(x
+
z)y

c)
F(x, y, z)
=
x

d)
F(x, y, z)
=
x y