Fundamentals of Boolean Algebra: Theory and Applications

1. Combinational logic --- outputs logical functions of inputs --- new outputs appear shortly after changed inputs (propagation delay) --- no feedback loops --- no clock Sequential logic --- outputs logical functions of inputs and previous history of circuit (memory) --- after changed inputs, new outputs appear in the next clock cycle --- frequent feedback loops

2. Fundamentals of Boolean algebra • Named after George Boole • He presented an algebraic formulation of the process of “logical thought and reason” • This formulation come to be known as Boolean Algebra

3. Postulates of Boolean algebra • Definition • A Boolean algebra is a closed algebraic system containing a set K of two or more elements and the two operators ‘•’ or ‘/\’ or ‘’, called AND, and ‘+’ or ‘\/’ or ‘’, called OR; • Closed system: for every a and b in set K, a•b belongs to K and a+b belongs to K.

4. Postulates of Boolean algebra • Existence of 1 and 0 • There exist unique elements 1 (one) and 0 (zero) in set A" such that for every a in K a) a + 0 = a, b) a • 1 = a, • where 0 is the identity element for the + operation and 1 is the identity element for the • operation.

5. Postulates of Boolean algebra • Commutativity of the + and • operations • For every a and b in K a) a + b = b + a. b) a • b = b • a • Associativity of the + and operations • For every a, b, and c in K a) a + (b + c) = (a + b) + c. b) a • (b • c) = (a • b) • c.

6. Postulates of Boolean algebra • Distributivity of + over • and • over + • For every a, b, and c in K a) a + (b • c) = (a + b) • (a + c), b) a • (b + c) = (a • b) + (a • c). • Existence of the complement • For every a in K there exists a unique element called ā (complement of a) in K such that a) a + ā = 1. b) a • ā = 0.

7. Venn diagrams for the postulates • Operations on sets Sets  closed regions Sets correspond to elements Intersection  corresponds to • Union  corresponds to +

8. Venn diagrams for the postulates

9. Venn diagrams Examples of Venn diagrams

10. Venn diagrams a + b • c = (a + b) • (a + c)

11. Venn diagrams a + b • c = (a + b) • (a + c)

12. Boolean algebra • Duality • If an expression f(x1, x2, … xn, +, •, 0, 1) is valid, then f(x1, x2, … xn, •, +, 1, 0) obtained by interchanging + and •, 0 and 1 is also valid a • (b + c) = (a • b) + (a • c) a + (b • c) = (a + b) • (a + c) Postulates 2 – 6 are stated in dual form

13. Fundamental theorems of Boolean algebra • Prove part (b) by exchanging + with •, and use the dual form of the postulates

14. Fundamental theorems of Boolean algebra

15. Fundamental theorems of Boolean algebra a • ā = 0 [P6(b)] a + ā = 1 [P6(a)] Therefore, ā is the complement of a, and also a is the complement of ā. Because the complement of ā is unique, it must be equal to a.

16. Fundamental theorems of Boolean algebra

17. Fundamental theorems of Boolean algebra • Why a + ab = a a ab b

18. Fundamental theorems of Boolean algebra

19. Fundamental theorems of Boolean algebra

20. Fundamental theorems of Boolean algebra

21. Fundamental theorems of Boolean algebra

22. Fundamental theorems of Boolean algebra

23. Fundamental theorems of Boolean algebra • Example using DeMorgan’s theorem

24. Fundamental theorems of Boolean algebra

25. Boolean algebra postulates and theorems

26. Theorems • Proofs by perfect induction Proofs by exhaustion: Let variables assume all possible values and show validity of result in all cases

27. (a) Keep axioms handy Example: Show X + 0 = X (b) Elaborate cases: if X = 0, have X + 0 = 0 + 0 = 0 = X if X = 1, have X + 0 = 1 + 0 = 1 = X

28. More Theorems Can prove by exhaustion....but have more cases For distributive laws, T8 looks like ordinary algebra T8’ also true (swap operators, factor, swap back) T9, T10 for logic minimization - drop irrelevant terms

29. T9, T10, T11 for logic minimization - drop superfluous terms T9 (Covering): X + XY = X and X(X+Y)=X Proof: X + XY = X1 + XY = X(1+Y) = X1 = X X(X+Y) = (X+0)(X+Y) = X+(0Y) = X+0 = X T10 (Combining): XY + XY’ = X and (X + Y)  (X + Y’) = X Proof: XY + XY’ = X(Y + Y’) = X1 = X (X + Y)(X + Y’) = X + (YY’) = X + 0 = X T11 (Consensus): XY+X’Z+YZ = XY+X’Z and (X+Y)(X’+Z)(Y+Z)= (X+Y)(X’+Z) Proof: If YZ = 0 XY+X’Z+YZ = XY+X’Z+ 0 = XY+X’Z else Y = Z = 1 left side: XY+X’Z+YZ = something + YZ = something + 1 =1 right side: XY+X’Z = X + X’ = 1 So, in either case, XY+X’Z+YZ = XY+X’Z If Y+Z = 1 (X+Y)(X’+Z)(Y+Z)= (X+Y)(X’+Z)1= (X+Y)(X’+Z) else Y = Z = 0 left side: (X+Y)(X’+Z)(Y+Z)= something  (Y + Z) = something  0 = 0 right side: (X+Y)(X’+Z) = (X+0)(X’+0) = XX’ = 0 So, in either case, (X+Y)(X’+Z)(Y+Z)= (X+Y)(X’+Z)

30. X + (X×Y) = X (T9)X× (X + Y) = X (dual)(X× X) + (X× Y) = X (T8)X+ (X× Y) = X (T3¢) parentheses,operator precedence Duality • De Morgan’s Theorems: (X + Y)’ = X’  Y’ (X  Y)’ = X’ + Y’ • Dual: Swap 0 & 1, AND & OR, but leave variables unchanged • Result: Theorems still true • Why? • f(X, Y) = g(X, Y) • complement[f(X, Y)] = complement[g(X, Y)] • dual[f(X’, Y’)] = dual[g(X’, Y’)] • but X’, Y’ just dummy variables, replace with originals • Counterexample?X + X × Y = X (T9)X × X + Y = X (dual)X + Y = X (T3¢)!! error ?

31. N-variable Theorems • Prove via induction • Most important: DeMorgan theorems

32. DeMorgan Symbol Equivalence Bubble-pushing...

33. Likewise for OR