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| | Hello buddy. Allow me introduce myself. extended [http://freebusinesscollege.com//read_blog/198199/great-ideas-about-auto-repair-that-anyone-can-use auto warranty] I am Luther Aubrey. The preferred pastime for him and his [http://www.allstate.com/tools-and-resources/car-insurance/whats-in-a-new-car-warranty.aspx children] is to drive and now he is [http://www.popularmechanics.com/cars/how-to/repair/how-to-get-a-used-car-warranty-and-not-get-screwed-6654348 attempting] to make cash with it. Kansas is where her house is auto warranty but she needs to transfer because of her family [http://www.uniqueindia.org/groups/the-hidden-tips-and-secrets-to-auto-repair/ extended auto warranty] members. My occupation is a extended car warranty manufacturing and distribution officer and [https://www.chevrolet.com/owners/warranty.html I'm performing] pretty good financially.<br><br>Visit my web site: [http://Phobianetworks.com/ActivityFeed/MyProfile/tabid/1141/UserId/17821/Default.aspx extended car warranty] |
| {| class="wikitable" align=right
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| |- bgcolor="#ddeeff" align="center"
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| |colspan=3|'''Variable inputs'''
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| |colspan=2| '''Function values'''
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| |- bgcolor="#ddeeff" align="center"
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| | ''x'' || ''y'' || ''z'' || <math>xy \vee \bar{x}z \vee yz</math> || <math>xy \vee \bar{x}z</math>
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| |- bgcolor="#ddffdd" align="center"
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| | 0 || 0 || 0 || 0 || 0
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| |- bgcolor="#ddffdd" align="center"
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| | 0 || 0 || 1 || 1 || 1
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| |- bgcolor="#ddffdd" align="center"
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| | 0 || 1 || 0 || 0 || 0
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| |- bgcolor="#ddffdd" align="center"
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| | 0 || 1 || 1 || 1 || 1
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| | 1 || 0 || 0 || 0 || 0
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| | 1 || 0 || 1 || 0 || 0
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| | 1 || 1 || 0 || 1 || 1
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| |- bgcolor="#ddffdd" align="center"
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| | 1 || 1 || 1 || 1 || 1
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| |}
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| In [[Boolean algebra (logic)|Boolean algebra]], the '''consensus theorem''' or '''rule of consensus'''<ref>Frank Markham Brown, ''Boolean Reasoning: The Logic of Boolean Equations'', 2nd edition 2003, p. 44</ref> is the identity:
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| :<math>xy \vee \bar{x}z \vee yz = xy \vee \bar{x}z</math>
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| The '''consensus''' or '''resolvent''' of the terms <math>xy</math> and <math>\bar{x}z</math> is <math>yz</math>. It is the conjunction of all the unique literals of the terms, excluding the literal which appears unnegated in one term and negated in the other.
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| The conjunctive [[De Morgan's laws|dual]] of this equation is:
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| :<math>(x \vee y)(\bar{x} \vee z)(y \vee z) = (x \vee y)(\bar{x} \vee z)</math>
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| ==Proof==
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| [[Sides of an equation|LHS]] = <math>xy \vee \bar{x}z \vee (x \vee \bar{x})yz</math>
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| = <math>xy \vee \bar{x}z \vee xyz \vee \bar{x}yz</math>
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| = <math>xy \vee xyz \vee \bar{x}z \vee \bar{x}yz</math>
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| = <math>xy(1 \vee z) \vee \bar{x}z(1 \vee y)</math>
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| = <math>xy \vee \bar{x}z</math>
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| = RHS
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| ==Consensus==
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| {{anchor|Consensus}}{{anchor|Opposition}}
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| The '''consensus''' or '''consensus term''' of two conjunctive terms of a disjunction is defined when one term contains the literal <math>a</math> and the other the literal <math>\bar{a}</math>, an '''opposition'''. The consensus is the conjunction of the two terms, omitting both <math>a</math> and <math>\bar{a}</math>, and repeated literals; the consensus is undefined if there is more than one opposition. For example, the consensus of <math>\bar{x}yz</math> and <math>w\bar{y}z</math> is <math>w\bar{x}z</math>.<ref>Frank Markham Brown, ''Boolean Reasoning: The Logic of Boolean Equations'', 2nd edition 2003, p. 81</ref>
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| The consensus can be derived from <math>(x\vee y)</math> and <math>(\bar{x} \vee z)</math> through the [[resolution (logic)|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).
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| ==Digital logic circuitry==
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| In [[digital logic]], including the consensus term in a circuit can eliminate [[race hazard]]s.
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| ==History==
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| The concept of consensus was introduced by Archie Blake in 1937.<ref name="blake">"Canonical expressions in Boolean algebra", Dissertation, Dept. of Mathematics, U. of Chicago, 1937, reviewed in J. C. C. McKinsey, ''The Journal of Symbolic Logic'' '''3''':2:93 (June 1938) {{DOI|10.2307/2267634}} {{jstor|2267634}}</ref> It was rediscovered by Samson and Mills in 1954<ref>Edward W. Samson, Burton E. Mills, Air Force Cambridge Research Center Technical Report 54-21, April 1954</ref> and by [[Willard van Orman Quine|Quine]] in 1955.<ref>W.V. Quine, "The problem of simplifying truth functions", ''American Mathematical Monthly'' '''59''':521-531, 1952</ref> Quine coined the term 'consensus'. Robinson used it for clauses in 1965 as the basis of his "[[resolution (logic)|resolution principle]]".<ref>J. Alan Robinson, "A Machine-Oriented Logic Based on the Resolution Principle", ''Journal of the ACM'' '''12''':1: 23–41.</ref><ref>D.E. Knuth, ''The Art of Computer Programming'' '''4A''': ''Combinatorial Algorithms'', part 1, p. 539</ref>
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| ==Notes==
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| {{Reflist|1}}
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| == References ==
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| * Roth, Charles H. Jr. and Kinney, Larry L. (2004, 2010). "Fundamentals of Logic Design", 6th Ed., p. 66ff.
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| {{DEFAULTSORT:Consensus Theorem}}
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| [[Category:Boolean algebra]]
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| [[Category:Theorems in propositional logic]]
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