Boolean Algebra and Circuits Andrew Knoll Ted Bealin

Boolean Algebra George Boole Invented Boolean Algebra in 1850 Algebraic rules applied to logic Foundation of modern electronics

Basic Properties of Boolean Algebra Similar to Standard Algebra: – Values – Operators

Basic Properties of Boolean Algebra Values ⁻True or False Operators ⁻AND, OR, NOT

Basic Properties of Boolean Algebra Written Form: ⁻P AND Q:PQ ⁻P OR Q: P + Q ⁻NOT P P’

ABABA+BA’B’ TTTTFF TFFTFT FTFTTF FFFFTT Basic Properties of Boolean Algebra

PropertyExamples Commutativex + y = y + xxy= yx Associative(x+y) + z = x + (y + z)(xy)z = x(yz) Distributivex + (yx) = (x + y)(x + z)x(y + z) = (xy) + (xz) Identityx + 0 = xx(1) = x Complementx + x’ = 1x(x’) = 0 Basic Properties of Boolean Algebra

1.abc’ + ab’c’ 2.ac’b + ac’b’Commutative Property 3.ac’(b + b’)Distributive Property 4.ac’ (1)Compliment 5.ac’Identity Basic Properties of Boolean Algebra

Gates and Logic Networks

Sum of Products Form abc + ab’c + ab’c’+ a’b’c + a’b’c’ Minimization abcf(a,b,c)

Sum of Products Form Minimization abcf(a,b,c)

Sum of Products Form abc + ab’c + ab’c’+ a’b’c Minimization abcf(a,b,c)

Karnaugh Map b Minimization aa’ b11 b’ abf(a,b)

Minimization abab’a’b’a’b c11 c’1 ab + bc

Minimization abab’a’b’a’b cd11 cd’11 c’d’11 c’d11 bd + b’d’