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Saturday, 17 March 2018

Consensus Theorem with Examples

Consensus Theorem in Digital Electronics are a powerful pair of theorems used in algebraic simplification of logic functions. The main theorem and its complementary may be stated as:

             xy + yz + zx′ = xy + zx′                                     (2.21a)
                 (x + y)(y + z)(z + x′) = (x+ y)(z +x′)                 (2.21b)
(a) Proof of the main theorem: First, multiply the middle term yz with  to yield

                          LHS = xy + (x + x′)yz + zx′ = xy + xyz + x′yz x′z
                                  = xy(1 + z) + x′z(1 + y) = xy + xz = RHS

(b) Proof of the complementary theorem: To prove the complement of the consensus theorem, as before, consider the LHS term:

                               LHS =  (x+ y)(y +z)(z + x′) = (xy + xz + y + yz)(z + x′)
                                        =  xyz + xz + yz + yx′ + xz + x′yz
                                        = xz + yz + yx′
                                     
Now, we expand RHS, which yields

                              RHS =  (x+y)(z+x′) = xz + yz +x′y
                                                                       
                              LHS = RHS

Extended Form I of the Consensus Theorem

This theorem states that:
                                      x′y′ + y′z′ + z′x = x′y′ + z′x

Proof:  We have the LHS given by

                                     LHS = x′y′ + y′z′ + z′x

Adding x and x′ to the middle term yields
                x′y′ + x′y′z′ + xy′z′ +z′x = x′y′(1+ z′) + zx′(1+ y′) = x′y′ + z′x = RHS
Complementary of Extended Form I

                                               (x′ + y′)(y′ + z′)(z′ + x) = (x′ + y′)(z′ + x)

Extended Form II of the Consensus Theorem

This theorem states that:
                                     xy +yzw + zx′ = xy + zx

Proof: Proof follows the same steps as given above. It can also be noticed that the middle term can contain any number of terms; all of them will get eliminated as illustrated below:

                                     xy + yzwabcd..... + zx′ = xy + zx′

STRAIGHT SIMPLIFICATION  BY  USING  BASIC  RULES

We now perform logic simplifications using the laws are rules given in Sections 2.16 and 2.17.  Reduction using basic laws and rules is known as straight simplification. The examples to follow will illustrate the procedure.



Example 1: Simplify the Consensus theorem

(a) Proof of the main theorem: First, multiply the middle term yz with  to yield

                                LHS = xy + (x + x′)yz + zx′ = xy + xyz + x′yz + x′z
                                = xy(1 + z) + x′z(1 + y) = xy + xz  = RHS

(b). Proof of the complementary theorem: To prove the complement of the consensus theorem, as before, consider the LHS term. Thus

                               LHS =  (x + y)(y + z)(z + x′) = (xy + xz +yy + yz)(z + x′)
                                        = xyz + xz + yz + yzx′ = xz + yz

It can be seen that this kind of simplification is based on hunch and experience, and can be quite difficult when the expressions become complex. So, we make use of the Karnaugh-Map method or the Quine-McCluskey tabular method for logic simplification.

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