Consensus theorem

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, (y + z). It can be derived from (x+y) and (x' +z) through the resolution inference rule. This shows that the LHS is derivable from the RHS (if A → B then A → AB; replacing A with RHS and B with (y + z) ). The RHS can be derived from the LHS simply through the conjunction elimination inference rule. Since RHS → LHS and LHS → RHS (in propositional calculus), then LHS = RHS (in Boolean algebra).

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

References

• Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.