Boolean Algebra

1. Boolean Algebra

2. Basic Definitions Boolean algebra: set of elements, set of operators, and axioms Axioms: • Closure • Associative Law • Commutative Law • Identity Element • Inverse • Distributive Law

3. Axiomatic Definition of Boolean Algebra A set B with operators + and • 1) closure + and • 2) identity x+0 = 0+x = x x•1 = 1•x = x 3) commutative x + y = y + x x•y = y•x 4) distributive x•(y + z) = (x•y) + (x•z) x + (y•z) = (x + y)•(x + z) 5) for x Œ B there exist x’ Œ B (complement) x + x’ = 1 and x•x’ = 0 6) at least two element x,y Œ B such that x ≠ y

4. Boolean algebra requires • elements of the set B • rules of operation for + and • • they satisfy the six postulates • Two-Valued Boolean Algebra • B = {0,1} • AND, OR, NOT operations • check postulates

5. Basic Theorems of Boolean Algebra • Duality • interchange OR and AND • interchange 0 and 1 • eg • x•1 = x • x + 0 = x • see table 2-1 • operator precedence • () • NOT • AND • OR • Venn Diagrams

6. Boolean functions • consider the functions: F1 = x’yz’ F2 = z + x’y’ F3 = x’yz’ + x’z + xy’z F4 = x’y + y’z • show truth table (like table 2-2) • note: F3 = F4 • obtain F4 by manipulating F3

7. Algebraic Manipulation • literal ==> primed or unprimed variable • simplify (minimize number of literals) x’ + xy’ x(x’+y) xy’z + x’y’z + xz’ xy + x’z + yz (x + y)(x’ + z)(y + z)

8. Solution x’ + xy’ = x’1 + xy’ = x’(y + y’) + xy’ = x’y + x’y’ + xy’ = x’y + x’y’ + x’y’ + xy’ = x’(y + y’) + y’(x’ + x) = x’ + y’ x(x’+y) = xx’ + x y’ = 0 + xy’ = xy’ xy’z + x’y’z + xz’ = y’z(x + x’) + xz’ = y’z + xz’ xy + x’z + yz = xyz’ + xyz + x’y’z + x’yz + xyz + x’yz = xyz’ + xyz + x’y’z + x’yz (eliminate duplicates) = xy(z + z’) + x’z(y + y’) = xy + x’z (x + y)(x’ + z)(y + z) = (x + y)(x’ + z) (dual of previous example)

9. Canonical and Standard Forms minterms • how can we represent a 1 in the truth table?

10. Canonical and Standard Forms maxterms • how can we represent a 0 in the truth table?

11. Other Logic Operators