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CONSENSUS THEOREM

CONSENSUS THEOREM. Choopan Rattanapoka. Introduction to The Consensus Theorem. The consensus theorem is very useful in simplifying Boolean expressions. Given an expression of the form XY + X’Z + YZ then term YZ is redundant and can be eliminated to form the equivalent expression

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CONSENSUS THEOREM

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  1. CONSENSUS THEOREM ChoopanRattanapoka

  2. Introduction to The Consensus Theorem • The consensus theorem is very useful in simplifying Boolean expressions. • Given an expression of the form • XY + X’Z + YZ then term YZ is redundant and can be eliminated to form the equivalent expression • XY + X’Z • The eliminated term is referred to as the consensus term.

  3. Consensus Term • Given a pair of terms for which a variable appears in one term and the complement of that variable in another. • The consensus term is formed by multiplying the two original terms together, leaving out the selected variable and its complement. • Example : • ABand A’C, consensus is BC • ABD and B’DE’, consensus is (AD)(DE’)  ADE’

  4. The consensus theorem (1) • The consensus theorem can be stated as follows: XY + X’Z + YZ = XY + X’Z • Proof XY + X’Z + YZ  XY + X’Z + (X + X’)YZ  XY + X’Z + XYZ + X’YZ  (XY + XYZ) + (X’Z + X’YZ)  XY(1 + Z) + X’Z(1 + Y)  XY + X’Z

  5. The consensus theorem (2) • Example : Simplify this expression A’B’ + AC + BC’ + B’C + AB A’B’ + AC + BC’ + B’C + AB Ans :A’B’ + AC + BC

  6. The consensus theorem (3) • The dual form of the consensus theorem is (X + Y)(X’ + Z)(Y + Z) = (X + Y)(X’ + Z) • Example : (A + B + C’)(A + B + D’)(B + C + D’) • The Consensus of (A + B + C’) and (B + C + D’) is (A + B + D’) • Thus, we can eliminate the consensus term • Answer : (A + B + C’)(A + B + D’)

  7. Consensus Term Eliminating Order (1) • Attention • The final result obtained by application of the consensus theorem may depend on the order in which terms are eliminated. • Example : A’C’D + A’BD + BCD + ABC + ACD’  Eliminate BCD terms (consensus of A’BD , ABC)  A’C’D + A’BD + ABC + ACD’ (No more eliminated term.)

  8. Consensus Term Eliminating Order (2) • Same Example : A’C’D + A’BD + BCD + ABC + ACD’  Eliminate A’BD terms (consensus of A’C’D , BCD)  A’C’D + BCD + ABC + ACD’  Eliminate ABC terms (consensus of BCD, ACD’)  A’C’D + BCD + ACD’ (no more eliminated term)

  9. Trick to use consensus theorem • Sometimes it is impossible to directly reduce an expression to a minimum number of terms by simply eliminating terms. • It may be necessary to first add a term using the consensus theorem and then use the added term to eliminate other terms.

  10. Example • F = ABCD + B’CDE + A’B’ + BCE’ • Consensus of ABCD and B’CDE  ACDE • Consensus of A’B’ and BCE’  ACE’ • But none of them appear in the original expression. • However, if we first add the consensus ACDE to F • F = ABCD + B’CDE + A’B’ + BCE’ + ACDE • Consensus of ACDE and A’B’  B’CDE • Consensus of ACDE and BCE’  ABCD • Thus, F = A’B’ + BCE’ + ACDE

  11. Exercise 1 • Simplify each of the following expressions using only the consensus theorem • BC’D’ + ABC’ + AC’D + AB’D + A’BD’ (reduce to 3 terms) • W’Y’ + WYZ + XY’Z + WX’Y (reduce to 3 terms)

  12. Algebraic Simplification (1) • Combining terms • XY + XY’ = X • Example : abc’d’ + abcd’ = abd’ (X = abd’, Y = d) • Complex example : • ab’c + abc + a’bc (X + X = X) • ab’c + abc+ abc+ a’bc • ac + bc

  13. Algebraic Simplification (2) • Eliminating terms • X + XY = X • Example : • a’b + a’bc = a’b (X = a’b, Y = c) • XY + X’Z + YZ = XY + X’Z (consensus theorem) • Example : • a’bc’ + bcd + a’bd = a’bc’ + bcd (X = c, Y = bd, Z = a’b)

  14. Algebraic Simplification (3) • Eliminating literals • X + X’Y = X + Y • Simply factoring may be necessary before the theorem is applied • Example : • A’B + A’B’C’D’ + ABCD’ = A’(B + B’C’D’) + ABCD’ = A’(B + C’D’) + ABCD’ = A’B + AC’D’ + ABCD’ = B(A’ + ACD’) + AC’D’ = B(A’ + CD’) + AC’D’ = A’B + BCD’ + AC’D’

  15. Algebraic Simplification (4) • Adding redundant terms. • Redundant terms can be introduced in several ways such as • adding xx’ • multiplying by (x + x’) • Adding yz to xy+x’z • Adding xy to x • Example : • WX + XY + X’Z’ + WY’Z’ = A’(B + B’C’D’) + ABCD’ = WX + XY + X’Z’ + WY’Z’ + WZ’ (add WZ’ by consensus term) = WX + XY + X’Z’ + WZ’ (WZ’ + WY’Z’  WZ’) = WX + XY + X’Z’ (eliminate WZ’ [consensus of WX and X’Z’])

  16. TODO • Simplify to a sum of three terms: • A’C’D’ + AC’ + BCD + A’CD’ + A’BC + AB’C’ • A’B’C’ + ABD + A’C + A’CD’ + AC’D + AB’C’

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