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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 XY + X’Z
The eliminated term is referred to as the consensus term.
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 : AB and A’C , consensus is BC ABD and B’DE’, consensus is (AD)(DE’)
ADE’
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
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
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’)
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.)
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)
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.
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
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)
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
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)
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’
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’])
TODO
Simplify to a sum of three terms:1) A’C’D’ + AC’ + BCD + A’CD’ + A’BC +
AB’C’2) A’B’C’ + ABD + A’C + A’CD’ + AC’D +
AB’C’