Chapter 4
Page 66
 Associativity of AND and OR is shown in Tables 43 and 44 in the book.

AND and OR identity values
x x ∧ 1 0
0
1
1
x x ∨ 0 0
0
1
1

AND and OR commutativity
x y x∧y y∧x 0
0
0
0
0
1
0
0
} 1
0
0
0
1
1
1
1
x y x∨y y∨x 0
0
0
0
0
1
1
1
} 1
0
1
1
1
1
1
1

AND distributive over OR
x y z (y∨z) (x∧y) (x∧z) (x∧(y∨z) (x∧y)∨(x∧z) 0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
1
0
1
0
0
0
0
0
1
1
1
0
0
0
0
1
0
0
0
0
0
0
0
1
0
1
1
0
1
1
1
1
1
0
1
1
0
1
1
1
1
1
1
1
1
1
1

AND annulment
x x ∧ 0 0
0
1
0

Involution of NOT
x (¬x) ¬(¬x) 0
1
0
1
0
1

OR distributive over AND is shown in Table 45 in book.

OR annulment value
x x ∨ 1 0
1
1
1

AND and OR complements
x ¬x x ∧ ¬x 0
1
0
1
0
0
x ¬x x ∨ ¬x 0
1
1
1
0
1

AND and OR idempotency
x x∧x 0
0
1
1
x x∨x 0
0
1
1

De Morgan’s law
x y (x∧y) ¬(x∧y) ¬x ¬y ¬x ∨ ¬y 0
0
0
1
1
1
1
0
1
0
1
1
0
1
1
0
0
1
0
1
1
1
1
1
0
0
0
0
x y (x∨y) ¬(x∨y) ¬x ¬y ¬x ∧ ¬y 0
0
0
1
1
1
1
0
1
1
0
1
0
0
1
0
1
0
0
1
0
1
1
1
0
0
0
0
Page 87

Let a 4bit integer be wxyz where each literal represents one bit. The even 4bit integers are given by the function:
F(w,x,y,z) = ¬w ∧ ¬x ∧ ¬y ∧ ¬z ∨ ¬w ∧ ¬x ∧ y ∧ ¬z ∨ ¬w ∧ x ∧ ¬ y ∧ ¬z ∨ ¬w ∧ x ∧ y ∧ ¬z ∨ w ∧ ¬x ∧ ¬y ∧ ¬z ∨ w ∧ ¬x ∧ y ∧ ¬z ∨ w ∧ x ∧ ¬y ∧ ¬z ∨ w ∧ x ∧ y ∧ ¬z
Using the distributive property repeatedly we get:
F(w,x,y,z) = ¬z ∧ (¬w ∧ ¬x ∧ ¬y ∨ ¬w ∧ ¬x ∧ y ∨ ¬w ∧ x ∧ ¬ y ∨ ¬w ∧ x ∧ y ∨ w ∧ ¬x ∧ ¬y ∨ w ∧ ¬x ∧ y ∨ w ∧ x ∧ ¬y ∨ w ∧ x ∧ y)
= ¬z ∧ (¬w ∧ (¬x ∧ ¬y ∨ ¬x ∧ y ∨ x ∧ ¬ y ∨ x ∧ y) ∨ (¬w ∧ (¬x ∧ ¬y ∨ ¬x ∧ y ∨ x ∧ ¬ y ∨ x ∧ y)
= ¬z ∧ (¬w ∨ w) ∧ (¬x ∧ ¬y ∨ ¬x ∧ y ∨ x ∧ ¬ y ∨ x ∧ y)
= ¬z ∧ (¬w ∨ w) ∧ (¬x ∧ (¬y ∨ y) ∨ x ∧ (¬ y ∨ y)
= ¬z ∧ (¬w ∨ w) ∧ (¬x ∨ x) ∧ (¬ y ∨ y)
And from the complement property we arrive at a minimal sum of products:
F(w,x,y,z) = ¬z
a simple NOT gate with the least significant bit as input.