Proving the Equivalence of Boolean Equations - III

 

Prove:

Majority = ((~A && B && C) || (A && ~B && C) ||

(A && B && ~C) || (A && B && C))

A
B
C
(~A && B && C)
(A && ~B && C)
(A && B && ~C)
(A && B && C)
Majority
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
0
0
0
0
0
0 1 1 1 0 0 0 1
1 0 0 0 0 0 0 0
1 0 1 0 1 0 0 1
1 1 0 0 0 1 0 1
1
1
1
0
0
0
1
1


Look at the equation:

((~A && B && C) || (A && ~B && C) || (A && B && ~C) || (A && B && C))

The first three groups contains two "normal" variables and the NOT of the other -- i.e., each corresponds to a case of having two true values. The fourth term corresponds to the case where all three are true. What is the significance of the NOT terms?

 

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Last Updated 02/14/2000
© L.Heath, 2000, updated by J.A.N. Lee, 2000/02/10