# UNIT 3 BOOLEAN ALGEBRA (CONT D)

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1 UNIT 3 BOOLEAN ALGEBRA (CONT D) Spring 2011

2 Boolean Algebra (cont d) 2 Contents Multiplying out and factoring expressions Exclusive-OR and Exclusive-NOR operations The consensus theorem Summary of algebraic simplification Proving validity of an equation Reading Unit 3

3 Objectives 3 In this unit, you will Continue to learn Boolean algebra Simplify, complement, multiply out and factor an expression Prove any theorem in an algebraic way or using a truth table Learn XOR

4 Guidelines for Multiplying Out and Factoring 4 Use 1. X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y For multiplying out, apply 2. and 3. before 1. to avoid unnecessary terms For factoring, apply 1., 2., 3. from right terms to left terms

5 Proof? 5 1. (X + Y)(X + Z) = X + YZ (X + Y)(X + Z) = XX + XZ + YX + YZ = X + XZ + XY + YZ = X + XY + YZ = X + YZ Or, X = 0, (0 + Y)(0 + Z) = YZ 0 + YZ = YZ X = 1, (1 + Y)(1 + Z) = 1 1 = YZ = 1 2. (X + Y)(X' + Z) = XZ + X'Y (X + Y)(X' + Z) = XX' + XZ + YX' + YZ = 0 + XZ + X'Y + YZ = XZ + X'Y + (X' + X)YZ = XZ + X'Y + X'YZ + XYZ expand & simplify = XZ(1 + Y) + X'Y(1 + Z) = XZ + X'Y

6 6 Multiplying Out Multiplying out: POS SOP Example: (A + B + C')(A + B + D)(A + B + E)(A + D' + E)(A' + C) = (A + B + C'D)(A + B + E)[AC + A'(D' + E)] 2. = (A + B + C'DE)(AC + A'D' + A'E) = AC + ABC + A'BD' + A'BE + A'C'DE = AC + A'BD' + A'BE + A'C'DE 1. X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y By brute force 162 terms!

7 7 Factoring Factoring: SOP POS Example: AC + A'BD' + A'BE + A'C'DE = A C + A'(BD' + BE + C'DE) 3. = X Z + X' Y = (X + Y)(X' + Z) = (A + BD' + BE + C'DE)(A' + C) = [A + C'DE + B(D' + E)](A' + C) 2. = [ X + Y Z ](A' + C) = (X + Y)(X + Z)(A' + C) = (A + C'DE + B)(A + C'DE + D' + E)(A' + C) X(Y + Z) = XY + XZ 2. (X + Y)(X + Z) = X + YZ 3. (X + Y)(X' + Z) = XZ + X'Y = (A + B + C')(A + B + D)(A + B + E)(A + D' + E)(A' + C)

8 8 XOR and XNOR Difference and equivalence

9 9 Operation Exclusive-OR Exclusive-OR (XOR) ( ) 0 0 = 0, 0 1 = 1, 1 0 = 1, 1 1 = 0 Symbol X Y X Y Truth table X Y X Y check difference of inputs Boolean expression: X Y = X'Y + XY' show the condition to make output ==1

10 Exclusive-OR Operations 10 X Y = X'Y + XY' Useful theorems: X 0 = X X 1 = X' X X = 0 X Y = Y X (commutative) (X Y) Z = X (Y Z) = X Y Z (associative) X(Y Z) = XY XZ (distributive) X X' = 1 (X Y)' = X Y' = X' Y = X'Y' + XY Prove distributive law? XY XZ = XY(XZ)' + (XY)'XZ (by definition) = XY(X' + Z') + (X' + Y')XZ (DeMorgan) = XYZ' + XY'Z = X(YZ' + Y'Z) = X(Y Z) (difference)' == equivalence

11 Operation Exclusive-NOR 11 Boolean expression: X Y = X'Y' + XY Exclusive-NOR (XNOR) ( ) check 0 0 = 1, 0 1 = 0, 1 0 = 0, 1 1 = 1 equivalence Symbol Truth table of inputs X X Y X Y (X Y) X Y Y X (X Y)' = (X Y) (difference)' == Y equivalence show the condition to make output ==1

12 12 Simplification of XOR and XNOR X Y = X'Y + XY' X Y = X'Y' + XY (X'Y + XY')' = X'Y' + XY e.g., F = (A'B C) + (B AC') = [A'BC + (A'B)'C'] +[B'AC' + B(AC')'] = A'BC + (A + B')C' + B'AC' + B(A' + C) = B(A'C + A' + C) + C'(A + B' + AB') = B(A' + C) + C'(A + B') (can be further simplified!) e.g., F = A' B C = (A'B' + AB) C = (A'B' + AB)C' + (A'B' + AB)'C = A'B'C' + ABC' + A'BC + AB'C

13 13 The Consensus Theorem Redundancy removal

14 The Consensus Theorem 14 XY + X'Z + YZ = XY + X'Z (YZ is redundant) Proof: XY + X'Z + YZ = XY + X'Z + (X + X')YZ = (XY + XYZ) + (X'Z + X'YZ) = XY(1 + Z) + X'Z(1 + Y) = XY + X'Z How to find consensus terms? 1. Find a pair of terms, one of which contains a variable and the other contains its complement A'C'D + A'BD + BCD + ABC + ACD' (A A') 2. Ignore the variable and its complement; the left variables composite the consensus term (A'BD ) + (ABC) BD BC = BCD (redundant term) e.g., a'b'+ ac + bc' + b'c + ab = a'b'+ ac + bc'

15 Ordering Does Matter! 15 e.g., A'C'D + A'BD + BCD + ABC + ACD' A'C'D + A'BD + BCD + ABC + ACD' = A'C'D + A'BD + ABC + ACD' 4 terms A'C'D + A'BD + BCD + ABC + ACD' = A'C'D + BCD + ACD' Only 3 terms!

16 Dual Form of the Consensus Theorem 16 XY + X'Z + YZ = XY + X'Z YZ is redundant dual (X + Y)(X' + Z)(Y + Z) = (X + Y)(X' + Z) (Y + Z) is redundant e.g., (a + b + c')(a + b + d')(b + c + d') = (a + b + c')(b + c + d') (a + b + c') + (b + c + d') a + b + b + d' = a + b + d'

17 Redundancy Injection 17 e.g., ABCD + B'CDE + A'B' + BCE' Find consensus terms ABCD + B'CDE ACDE ABCD + A'B' BCD(B') + (ACD)A' B'CDE + BCE' CDE(CE') + B'CD(BC) A'B' + BCE' A'CE' Add consensus term ACDE ABCD + B'CDE + A'B' + BCE' + ACDE = A'B' + BCE' + ACDE

18 18 Summary of Algebraic Simplification Simplification can reduce cost

19 Rule A -- Combining Terms 19 XY + XY' = X(Y + Y') = X e.g., abc'd' + abcd' = abd' (X = abd', Y = c) ab'c + abc + a'bc = ab'c + abc + abc + a'bc (repeat term) = ac + bc (a + bc)(d + e') + a'(b' + c')(d + e') = d + e' (why?? DeMorgan)

20 20 Rule B -- Eliminating Terms 1. X + XY = X (keep the boss) 2. XY + X'Z + YZ = XY + X'Z (consensus) e.g., a'b+ a'bc = a'b (X = a'b) a'bc' + bcd + a'bd = a'bc'+ bcd (X = c, Y = bd, Z = a'b)

21 Rule C -- Eliminating Literals 21 X + X'Y = (X + X')(X + Y) = X + Y e.g., A'B + A'B'C'D' + ABCD' = A'(B + B'C'D') + ABCD' (common term: A') = A'(B + C'D') + ABCD' (Rule C) = A'B + A'C'D' + ABCD' = B(A' + ACD') + A'C'D' (common term: B) = B(A' + CD') + A'C'D' (Rule C) = A'B + BCD' + A'C'D' (final terms)

22 Rule D -- Adding Redundant Terms Y = Y + XX' 2. Y = Y(X + X') 3. XY + X'Z = XY + X'Z + YZ 4. X = X + XY e.g., Add redundancy to eliminate other terms WX + XY + X'Z' + WY'Z' = WX + XY + X'Z' + WY'Z' + WZ' (add WZ' by consensus thm) = WX + XY + X'Z' + WZ' (eliminate WY'Z' by WZ') = WX + XY + X'Z' (consensus again)

23 Another Example 23 A'B'C'D' + A'BC'D' + A'BD + A'BC'D + ABCD + ACD' + B'CD' = (Apply rules A, B, C, D) = A'C'D' + A'BD + B'CD' + ABC No easy way to determine when a Boolean expression has a minimum # of terms or literals Systematic way will be discussed in Unit 5 & Unit 6

24 24 Proving Validity of an Equation

25 How to Determine if an Equation Valid? Construct a truth table (proof by cases) 2. Manipulate one side until it is identical to the other side 3. Reduce both sides independently to the same expression 4. Perform the same operation on both sides if the operation is reversible (Boolean algebra ordinary algebra) Complement is reversible Multiplication/division and addition/subtraction are not reversible x+y = x+z does not imply y=z (e.g., x=1, y=0, z=1) xy = xz does not imply y=z (e.g., x=0, y=0, z=1) Using 2. and 3., usually 1. Reduce both sides to (minimum) SOP or POS 2. Compare both sides 3. Try to add or delete terms by using theorems

26 Example 26 Show that A'BD' + BCD + ABC' + AB'D = BC'D' + AD + A'BC By the consensus theorem, A'BD' + BCD + ABC' + AB'D = A'BD' + BCD + ABC' + AB'D + A'BC + BC'D' + ABD = AD + A'BD' + BCD + ABC' + A'BC + BC'D' = AD + A'BC + BC'D'

27 One More Example 27 Show that A'BC'D + (A' + BC)(A + C'D') + BC'D + A'BC' = ABCD + A'C'D' + ABD + ABCD' + BC'D 1. Reduce the left side: A'BC'D + (A' + BC)(A + C'D') + BC'D + A'BC' = (A' + BC)(A + C'D') + BC'D + A'BC' = A'C'D' + ABC + BC'D + A'BC' (multiplying out) = A'C'D' + ABC + BC'D (consensus) 2. Reduce the right side: ABCD + A'C'D' + ABD + ABCD' + BC'D = ABC + A'C'D' + ABD + BC'D = ABC + A'C'D' + BC'D (consensus) 3. Because both sides were independently reduced to the same expression, the original equation is valid.

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