1. CONSENSUS THEOREM
Choopan Rattanapoka

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 :
AB and 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
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)
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’