Section 4.4

Section 4.4
Boolean Theorems

Investigating the various Boolean theorems (rules) can help us to simplify logic expressions and logic circuits.

Multivariable Theorems

The theorems presented below involve more than one variable:

(9) x + y = y + x (commutative law)

(10) x * y = y * x (commutative law)

(11) x+ (y+z) = (x+y) +z = x+y+z (associative law)

(12) x (yz) = (xy) z = xyz (associative law)

(13a) x (y+z) = xy + xz

(13b) (w+x)(y+z) = wy + xy + wz + xz

(14) x + xy = x [proof see below]

(15) x + x'y = x + y

Proof of (14)

x + xy = x (1+y)

= x * 1 [using theorem (6)]

= x [using theorem (2)]

(9)	x + y = y + x (commutative law)
(10)	x * y = y * x (commutative law)
(11)	x+ (y+z) = (x+y) +z = x+y+z (associative law)
(12)	x (yz) = (xy) z = xyz (associative law)
(13a)	x (y+z) = xy + xz
(13b)	(w+x)(y+z) = wy + xy + wz + xz
(14)	x + xy = x [proof see below]
(15)	x + x'y = x + y

x + xy	= x (1+y)
	= x * 1 [using theorem (6)]
	= x [using theorem (2)]