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Unit 3: Boolean Algebra

3.2.1 Multivariable Theorems Notes
The theorems presented below involve more than one variable:

(9) x + y = y + x
(10) x • y = y • x
(11) x + (y + z) = (x + y) + z = x + y + z
(12) x(yz) = (xy)z = xyz

(13a) x(y + z) = xy + xz
(13b) (w + x)(y + z) = wy + xy + wz + xz
(14) x + xy = x
–
(15) x + xy = x + y
Theorems (9) and (10) are called the commutative laws. These laws indicate that the order in
which we OR Or AND two variables is unimportant; the result is the same.
Theorems (11) and (12) are the associative laws, which state that we can group the variables in
an AND expression or OR expression any way we want.
Theorem (13) is the distributive law, which states that an expression can be expanded by
multiplying term by term just the same as in ordinary algebra. This theorem also indicates that we
can factor an expression. That is, if we have a sum of two (or more) terms, each of which contains
a common variable, the common variable can be factored out just like in ordinary algebra. For
example, if we have the expression ABC + ABC, we can factor out the B variable:
ABC + ABC = BAC( + AC)

As another example, consider the expression ABC + ABD. Here the two terms have the variables
A and B in common, and so A• B can be factored out of both terms. That is,
ABC + ABD = AB(C + D)

Theorems (9) to (13) are easy to remember and use since they are identical to those of ordinary
algebra. Theorems (14) and (15), on the other hand, do not have any counterparts in ordinary
algebra. Each can be proved by trying all possible cases for x and y. This is illustrated for theorem
(14) as follows:
Case 1. For x = 0, y = 0,
x + xy = x

0 + 0 • 0 = 0
0 = 0
Case 2. For x = 0, y = 1,
x + xy = x
0 + 0 • 1 = 0
0 + 0 = 0
0 = 0
Case 3. For x = l, y = 0,
x + xy = x
1 + 1 • 0 = 1
1 + 0 = 1
1 = 1

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