View
224
Download
0
Category
Preview:
Citation preview
Simplifying Boolean Expressions
Boolean Operators (T/F)x y x AND y
F F F
F T F
T F F
T T T
x y x OR y
F F F
F T T
T F T
T T T
x y x XOR y
F F F
F T T
T F T
T T F
x NOT x
F T
T F
Boolean Operators (1/0)x y x AND y
0 0 0
0 1 0
1 0 0
1 1 1
x y x OR y
0 0 0
0 1 1
1 0 1
1 1 1
x y x XOR y
0 0 0
0 1 1
1 0 1
1 1 0
x NOT x
0 1
1 0
Boolean Operators Symbols
Operator Symbol
NOT ā (overbar), a’, ~a
AND · (mult. dot)
OR +
XOR (plus sign with (plus sign with circle around it)circle around it)
Simplifying Boolean Expressions Commutative laws
A + B = B + AA · B = B · A
Associative lawsA + (B + C) = (A + B) + CA · (B · C) = (A · B) · C
Distributive lawsA · (B + C) = A · B + A · CA + (B · C) = (A + B) · (A + C)
Simplifying Boolean Expressions Tautology laws
A · A = AA + A = AA + ~A = 1A · ~A = 0
Absorption LawA + (A · B) = AA · (A + B) = A
Simplifying Boolean Expressions
Identities0 · A = 0
0 + A = A
A + 1 = 1
1 · A = A
A = A
ComplementA + ~A · B = A + B
Examples A + A + A + A = A
Using the Tautology law
A Bigger ExampleSimplify ~A · B + A · ~B + ~A · ~B
~A · B + A · ~B + ~A · ~B ~A · B + (A · ~B + ~A · ~B) Associative~A · B + (~B · (A + ~A)) Distributive~A · B + ~B & Tautology~A + ~B Complement
Verify with a truth table!
Practice Show that A + B · C = (A + B) · (A + C) is
true using a truth table.
Practice Show that A + ~A · B = A + B
Practice Simplification Simplify A + AB + ~B and verify with a truth
table
De Morgan’s Laws~(A · B) = ~A + ~B~A · ~B = ~(A+B)
1. Take a term ~A · ~B
2. NOT the individual members of the term A · B
3. Change the operator i.e. · to +, or + to ·A + B
4. NOT the entire term~(A+B)
De Morgan’s Law Example
f = ~A · ~B + (~A + ~B)
= ~~( ~A · ~B + (~A + ~B) ) NOT NOT
= ~( (A + B) · ~(~A + ~B) ) De Morgan’s
= ~( (A + B) · (A·B) ) De Morgan’s
= ~( A·(A·B) + B·(A·B) ) Distributive
= ~( A·B + A·B ) Tautology
= ~(A·B) Tautology
Recommended