Consensus theorem

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Variable inputs Function values
X Y Z xy + x'z + yz xy + x'z
0 0 0 0 0
0 0 1 1 1
0 1 0 0 0
0 1 1 1 1
1 0 0 0 0
1 0 1 0 0
1 1 0 1 1
1 1 1 1 1

In Boolean algebra, the consensus theorem is a simplification of the following terms:

xy + x'z + yz = xy + x'z

Proof for this theorem is:

   LHS = 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
       = RHS

The dual of this equation is:

(x + y)(x' + z)(y + z) = (x + y)(x' + z)

The consensus term refers to the redundant term.

In digital logic, including the consensus term can eliminate race hazards.

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