en>PJTraill: /* See also */ Graded (mathematics)

2014-02-04T01:01:07Z

← Older revision		Revision as of 03:01, 4 February 2014
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	~~Irwin Butts~~ is ~~what my spouse loves~~ to ~~call me although I don~~'~~t truly like being known as like~~ that. ~~Minnesota has usually been his house but his spouse desires them~~ to ~~move~~. ~~I am~~ a ~~meter reader~~. The ~~preferred hobby~~ for ~~my children and me~~ is to ~~perform baseball~~ and I'~~m attempting to make it a occupation~~.<br><br>~~my site:~~ [http://~~Bonetoob~~.~~com~~/~~blog~~/~~432777~~ http://~~Bonetoob~~.~~com~~/~~blog~~/~~432777~~]		In [[Boolean logic]], an '''implicant''' is a "covering" (sum term or product term) of one or more [[minterms]] in a [[sum of products]] (or [[maxterms]] in a [[product of sums]]) of a boolean function. Formally, a [[product term]] ''P'' in a [[sum of products]] is an '''implicant''' of the [[Boolean function]] ''F'' if ''P'' [[Logical implication\|implies]] ''F''. More precisely:

			: ''P'' implies ''F'' (and thus is an implicant of ''F'') if ''F'' also takes the value 1 whenever ''P'' equals 1.
			where
			* ''F'' is a [[Boolean function]] of ''n'' variables.
			* ''P'' is a [[product term]].

			This means that P<math>=></math>F with respect to the natural ordering of the Boolean space. For instance, the function

			:<math>f(x,y,z,w)=xy+yz+w</math>

			is implied by <math>xy</math>, by <math>xyz</math>, by <math>xyzw</math>, by <math>w</math> and many others; these are the implicants of <math>f</math>.
			== Prime implicant ==
			A '''prime implicant''' of a function is an implicant that cannot be covered by a more general (more reduced - meaning with fewer [[literal (computer programming)\|literals]]) implicant. [[W.V. Quine]] defined a ''prime implicant'' of ''F'' to be an implicant that is minimal - that is, the removal of any literal from ''P'' results in a non-implicant for ''F''. '''Essential prime implicants''' are prime implicants that cover an output of the function that no combination of other prime implicants is able to cover.

			Using the example above, one can easily see that while <math>xy</math> (and others) is a prime implicant, <math>xyz</math> and <math>xyzw</math> are not. From the latter, multiple literals can be removed to make it prime:

			*<math>x</math>, <math>y</math> and <math>z</math> can be removed, yielding <math>w</math>.
			*Alternatively, <math>z</math> and <math>w</math> can be removed, yielding <math>xy</math>.
			*Finally, <math>x</math> and <math>w</math> can be removed, yielding <math>yz</math>.

			The process of removing literals from a Boolean term is called '''expanding''' the term. Expanding by one literal doubles the number of input combinations for which the term is true (in binary Boolean algebra). Using the example function above, we may expand <math>xyz</math> to <math>xy</math> or to <math>yz</math> without changing the cover of <math>f</math>. <ref>De Micheli, Giovanni. ''Synthesis and Optimization of Digital Circuits''. McGraw-Hill, Inc., 1994</ref>

			The sum of all prime implicants of a Boolean function is called its '''complete sum''', '''minimal covering sum''', or [[Blake canonical form]].

			==See also==
			* [[Quine–McCluskey algorithm]]
			* [[Karnaugh map]]

			==References==
			<references/>

			==External links==
			* [http://www.cs.ualberta.ca/~amaral/courses/329/webslides/Topic4-KarnaughMaps/sld021.htm Slides explaining implicants, prime implicants and essential prime implicants]
			* [http://web.cecs.pdx.edu/~mcnames/ECE171/Lectures/Lecture10.html Examples of finding essential prime implicants using K-map]

			[[Category:Boolean algebra]]

en>Gaius Cornelius: /* Linear maps */Fix indefinite article, replaced: an commutative → a commutative using AWB

2011-11-29T21:25:35Z

Linear maps: Fix indefinite article, replaced: an commutative → a commutative using AWB

New page

Irwin Butts is what my spouse loves to call me although I don't truly like being known as like that. Minnesota has usually been his house but his spouse desires them to move. I am a meter reader. The preferred hobby for my children and me is to perform baseball and I'm attempting to make it a occupation.<br><br>my site: [http://Bonetoob.com/blog/432777 http://Bonetoob.com/blog/432777]

Graded vector space - Revision history

en>PJTraill: /* See also */ Graded (mathematics)

en>Gaius Cornelius: /* Linear maps */Fix indefinite article, replaced: an commutative → a commutative using AWB