Formal Proofs: Boolean Algebra

Show that (x + x'y)' = x'y' using laws of Boolean Algebra

(x + x'y)' = x'y' 1. (x + x'y)' Given 2. x' * (x'y)' 1. de morgan's laws 3. x' * (x'' + y') 2. de morgan's laws 4. x' * (x + y') 3. Double complement law 5. x'x + x'y' 4. Distribution 6. 0 + x'y' 5. Complement 7. x'y' 6. Identity Law

Associative Laws

(x + y) + z = x + (y + z) (xy)z = x(yz)

De Morgan's Laws

(x + y)' = x'y' (xy)' = x' + y'

Double Complement Laws

(x')' = x

Show that x(y' + x) = x using laws of Boolean Algebra

1. x(y' + x) Given 2. xy' + xx 1. Distribution 3. xy' + x 2. Idempotent Laws 4. x (y' + 1) 3. Distribution 5. x (1) 4. Dominate 6. x 5. Identity

What operation is the OR operator?

Addition

What operation is the NOT operator?

Complement

Boolean Function Expression

F(p,q) = pq' = p * q'

What operation is the AND operator?

Multiplication

Literal

Single Boolean variable of its complement

Minterm

a product of literals for each variable (row from I/O table)

XOR

p or q but not p and q

Domination Laws

x * 0 = 0 x + 1 = 1

Absorption Laws

x + (xy) = x x(x + y) = x

Identity Laws

x + 0 = x x * 1 = x

Idempotent Laws

x + x = x x * x = x

Commutative Laws

x + y = y + x xy = yx

Distributive Laws

x + yz = (x + y)(x + z) x(y + z) = xy + xz

Complement Laws

xx' = 0 1' = 0 x + x' = 1 0' = 1