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6108:{\displaystyle {\begin{aligned}\lambda {}ab.baa&\Leftrightarrow a\operatorname {p_{1}} b\\\lambda {}ab.abb&\Leftrightarrow a\operatorname {p_{2}} b\\\lambda {}ab.aab\Leftrightarrow \lambda {}ab.aKb&\Leftrightarrow {\begin{cases}a\lor b&{\text{if }}K=1\\a\land b&{\text{if }}K=0\end{cases}}\\\lambda {}ab.aba\Leftrightarrow \lambda {}ab.abKi&\Leftrightarrow {\begin{cases}a\land b&{\text{if }}Ki=0\\a\lor b&{\text{if }}Ki=1\end{cases}}\\\lambda {}ab.abK&\Leftrightarrow {\begin{cases}a\rightarrow b&{\text{if }}K=1\\a{\overline {\leftarrow }}b&{\text{if }}K=0\end{cases}}\\\lambda {}ab.baK&\Leftrightarrow {\begin{cases}a\leftarrow b&{\text{if }}K=1\\a{\overline {\rightarrow }}b&{\text{if }}K=0\end{cases}}\\\lambda {}ab.aKib&\Leftrightarrow {\begin{cases}a{\overline {\leftarrow }}b&{\text{if }}Ki=0\\a\rightarrow b&{\text{if }}Ki=1\end{cases}}\\\lambda {}ab.bKia&\Leftrightarrow {\begin{cases}a{\overline {\rightarrow }}b&{\text{if }}Ki=0\\a\leftarrow b&{\text{if }}Ki=1\end{cases}}\end{aligned}}}
6184:, since the template pushed the table downwards and created a big blank space. As to whether it's useless, I think the template is useful as soon as any logical notation is featured, since it shows different notational variants for each of the connectives, so readers who are used to, say, & instead of ∧, will not be confused by the discrepancy between what they're used to and the article. But if you don't want it in the lead, again, I don't actually care enough to undo your edits, I did it by accident because I thought I must have forgotten to do it.
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5248:{\displaystyle {\begin{aligned}\lambda {}a.aKiKi&\Leftrightarrow {\begin{cases}\bot &{\text{if }}Ki=0\\\top &{\text{if }}Ki=1\end{cases}}\\\lambda {}a.aKK&\Leftrightarrow {\begin{cases}\top &{\text{if }}K=1\\\bot &{\text{if }}K=0\end{cases}}\\\lambda {}a.aKKi&\Leftrightarrow a\\\lambda {}a.aKiK&\Leftrightarrow \neg {}a\end{aligned}}}
4956:{\displaystyle {\begin{aligned}aKia\Leftrightarrow aKiKi&\Leftrightarrow Ki\\aaK\Leftrightarrow aKK&\Leftrightarrow K\\aaa\Leftrightarrow aaKi\Leftrightarrow aKa\Leftrightarrow aKKi&\Leftrightarrow a\\aKiK&\Leftrightarrow {\begin{cases}Ki&{\text{when }}a=K\\K&{\text{when }}a=Ki\end{cases}}\end{aligned}}}
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I scanned the rules here at this page, and my extremely formal and pedantic justification for doing this seems lacking. I'm not even sure this page has defined equality as an operator resolving to a truth value (as common in many computer languages). A better mathematician than I will spot whether
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The article suggests that
Boolean algebra uses the same notation as propositional logic, but this is incorrect. Modern Boolean algebra uses the same notation as elementary algebra (i.e., the dot symbol, plus sign, etc.). Could an established editor research with mainstream sources and update the
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The word "countable" does not appear in this article. Nor is it mentioned that some boolean algebras are not power sets. Never mind that there are (40-year-old) classification theorems for countable boolean algebras. I would love it if this was remedied, either in this article, or in some other.
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are equivalent structures, one using logical connectives as operators, and the other using arithmetic operations. You are asserting that
Boolean rings are more used in modern mathematics. This seems not true. If this were true, the
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This is
Boolean algebra as filtered through how an ancient CS guy conceptualizes symbolic reasoning. Pedantic we can do. My problem is to make the rule set complete, I seemed to need to add a problematic rule:
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Hi, I remade the edit as an accident, I didn't notice it was reverted. The logical connectives sidebar really does fit better in the "Operations" section semantically speaking, but I was moving it to the lead
950:): "Contrary to popular belief Boole did not work with a two-element Boolean algebra, nor with the Boolean algebra of subsets of a given set. Boole was simply not doing Boolean algebra, nor Boolean rings."
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To find the dual operator one has to negate the operands and find the operator that provides the opposite results with the negated operands. Here are the truth tables of all 8 dual operator pairs:
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Just to be clear, my strange rule reads to a CS guy as "rewrites as" at the level of whole equations. I'm well aware this is a frame jump, even if it also happens to be algebraically sound. —
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this is indeed an axiomatic gap at this page, or a merely an expository gap, or merely a gap of implied application, or even more merely the braino of an antique from across the aisle.
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1243:{\displaystyle {\begin{aligned}\neg {0}&=1\\\neg {1}&=0\\\neg {(\neg {x})}&=x\\(\neg {x}=y)&\rightarrow (x=\neg {y})\\x\vee 0&=x\\0\vee x&=x\\\end{aligned}}}
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Also, if you can standardize the pedagogy, fill your boots. I was trying to avoid the arithmetic metaphor for this slight application, though perhaps that was misguided.
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Note that I'm an extreme tumbleweed editor and I'm unlikely to return here; this is already far more time than I ordinarily expend in any one place. —
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Alas, I am not finding such an exposition in
Knowledge. I do not have the wherewithal to be bold and write such content myself. Anyone?
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If we bind the variables with lambda expressions, we get all four possible
Boolean unary operators - but not unambiguously.
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in the operator position merely select an argument. Almost half of the others are duplicates with renamed variables.
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related articles on
Knowledge. If you would like to participate, please visit the project page, where you can join
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follow from the definition above, the rules of negation, and the commutative constant identity for disjunction:
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This is problematic because it seems to hoist a level above the definition of rightarrow recently introduced.
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4523:{\displaystyle {\begin{aligned}KKKi&\Leftrightarrow K\\KiKKi&\Leftrightarrow Ki\end{aligned}}}
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6231:(the multiplication of Boolean rings) would be more used than the ordinary or. This is not the case.
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We get 8 of 16 Boolean binary operators but because of duality it is impossible to decide whether
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Suppose we try to figure out which combinator should be 0 and which one should be 1 by applying
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Lambda calculus is written in prefix-notation, so the leftmost symbol is the operator.
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Requested articles/Applied arts and sciences/Computer science, computing, and
Internet
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Let's have a look at the ones with just one of the variables (either a or b) first.
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and two variables to form such a binary function application. The 32 triplets with
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be a function that takes two arguments and returns the first (selector-1) and let
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https://www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PREPRINTS/aboole.pdf
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In lambda calculus for instance
Boolean values can be represented by the
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1409:{\textstyle 1=x\rightarrow y=1\rightarrow y=\neg {1}\vee y=0\vee y=y=1}
1724:, but this is a distinct axiom, and if it is rejected, then you get
1551:{\textstyle 1=x\rightarrow y=x\rightarrow 0=\neg {x}\vee 0=\neg {x}}
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begins to tackle this. Still, this article should mention this.
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and assorted other logics. Much of it is not straight-forward.
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See the current status of my new section on Boolean algebra at
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Now let's have a look at the expressions with both variables.
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Find pictures for the biographies of computer scientists (see
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This article is not about Boolean algebras — that's at
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be a function that returns the second (selector-2).
1677:{\displaystyle (\neg {x}=y)\rightarrow (x=\neg {y})}
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979:. For discussion, here's my addition just added:
652:Computer science articles needing expert attention
6182:because that section looked crowded in my monitor
33:for general discussion of the article's subject.
6276:Knowledge level-4 vital articles in Mathematics
3833:{\displaystyle {\overline {\leftrightarrow }}}
792:WikiProject Computer science/Unreferenced BLPs
923:This page has archives. Sections older than
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709:Computer science articles without infoboxes
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2896:{\displaystyle {\overline {\rightarrow }}}
1035:{\textstyle x\rightarrow y=\neg {x}\vee y}
613:Here are some tasks awaiting attention:
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2169:{\displaystyle {\overline {\land }}}
1772:Boolean algebras canonically defined
942:Boole did not invent Boolean Algebra
534:This article is within the scope of
445:This article is within the scope of
356:This article is within the scope of
3530:{\displaystyle {\overline {p_{2}}}}
2292:{\displaystyle {\overline {\lor }}}
1728:. From there, things snowball into
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127:Talk:Boolean algebra (structure)
45:Click here to start a new topic.
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808:Tag all relevant articles in
548:and see a list of open tasks.
459:and see a list of open tasks.
370:and see a list of open tasks.
42:Put new text under old text.
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2773:{\displaystyle \leftarrow }
1790:Boolean algebra (structure)
1750:Countable boolean algebras?
748:List of computer scientists
50:New to Knowledge? Welcome!
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1722:law of the excluded middle
580:project's importance scale
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6214:17:36, 21 July 2024 (UTC)
4156:{\displaystyle \neg {}b)}
4106:{\displaystyle (\neg {}a}
3862:{\displaystyle \neg {}b)}
3805:{\displaystyle (\neg {}a}
3559:{\displaystyle \neg {}b)}
3495:{\displaystyle (\neg {}a}
3244:{\displaystyle \neg {}b)}
3180:{\displaystyle (\neg {}a}
2925:{\displaystyle \neg {}b)}
2868:{\displaystyle (\neg {}a}
2624:{\displaystyle \neg {}b)}
2567:{\displaystyle (\neg {}a}
2321:{\displaystyle \neg {}b)}
2264:{\displaystyle (\neg {}a}
2013:{\displaystyle \neg {}b)}
1963:{\displaystyle (\neg {}a}
810:Category:Computer science
586:
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560:Computer science articles
522:
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433:
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80:Be welcoming to newcomers
812:and sub-categories with
402:project's priority scale
359:WikiProject Mathematics
6256:C-Class vital articles
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1868:{\displaystyle \land }
1848:
1798:propositional calculus
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773:Computer science stubs
448:WikiProject Statistics
75:avoid personal attacks
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4127:{\displaystyle \bot }
4108:
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4077:{\displaystyle \neg }
4058:
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4011:{\displaystyle \top }
3992:
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3776:{\displaystyle \neg }
3757:
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3466:{\displaystyle \neg }
3447:
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3400:{\displaystyle p_{2}}
3374:
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3151:{\displaystyle \neg }
3132:
3111:
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2839:{\displaystyle \neg }
2820:
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2538:{\displaystyle \neg }
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2235:{\displaystyle \neg }
2216:
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2143:
2015:
1986:
1984:{\displaystyle \lor }
1965:
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1934:{\displaystyle \neg }
1915:
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286:level-4 vital article
233:Auto-archiving period
100:Neutral point of view
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1321:{\displaystyle x=1,}
1303:
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989:material conditional
591:Things you can help
382:mathematics articles
105:No original research
6199:Incorrect notation?
1476:{\displaystyle y=0}
471:Statistics articles
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52:Learn to edit
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6229:exclusive or
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398:Top-priority
397:
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323:Top‑priority
301:WikiProjects
284:
236:
202:
189:
183:
175:
168:
162:
156:
150:
140:
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19:This is the
4269:Motivation:
1483:, we have:
1257:, assuming
373:Mathematics
364:mathematics
320:Mathematics
166:free images
31:not a forum
6250:Categories
462:Statistics
453:statistics
425:Statistics
1802:Trovatore
1794:mass noun
1328:we have:
897:Archive 5
892:Archive 4
885:Archive 3
880:Archive 2
875:Archive 1
671:Computing
289:is rated
88:if needed
71:Be polite
21:talk page
6233:D.Lazard
1295:implies
995:implies
860:Archives
719:Maintain
662:Copyedit
204:Archives
120:Shortcut
56:get help
29:This is
27:article.
1818:Duality
925:90 days
700:Infobox
638:Cleanup
578:on the
489:on the
400:on the
291:C-class
237:90 days
172:WP refs
160:scholar
1709:MaxEnt
1699:MaxEnt
987:, the
681:Expand
297:scale.
144:Google
4927:when
4904:when
868:Index
764:Stubs
738:Photo
595:with:
278:This
209:Index
187:JSTOR
148:books
84:Seek
6237:talk
6222:and
6210:talk
6190:talk
6166:talk
4595:and
4552:and
4293:and
1806:talk
1800:. --
1780:talk
1761:talk
1738:talk
1457:and
1419:For
1253:For
1048:and
958:talk
570:High
180:FENS
154:news
73:and
6137:or
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481:Top
392:Top
194:TWL
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4255:0
4252:0
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3104:)
3101:b
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2792:)
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2616:b
2562:a
2554:(
2491:)
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2251:(
2188:)
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2040:1
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2030:0
2027:0
2024:0
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2005:b
1958:a
1950:(
1887:)
1884:b
1842:a
1839:(
1804:(
1778:(
1759:(
1736:(
1672:)
1668:y
1661:=
1658:x
1655:(
1649:)
1646:y
1643:=
1639:x
1632:(
1604:0
1601:=
1598:x
1595:=
1592:)
1588:x
1581:(
1575:=
1571:1
1545:x
1538:=
1535:0
1528:x
1521:=
1518:0
1512:x
1509:=
1506:y
1500:x
1497:=
1494:1
1471:0
1468:=
1465:y
1445:1
1442:=
1438:y
1431:x
1404:1
1401:=
1398:y
1395:=
1392:y
1386:0
1383:=
1380:y
1373:1
1366:=
1363:y
1357:1
1354:=
1351:y
1345:x
1342:=
1339:1
1316:,
1313:1
1310:=
1307:x
1297:y
1293:x
1291:(
1279:1
1276:=
1272:y
1265:x
1234:x
1231:=
1224:x
1218:0
1211:x
1208:=
1201:0
1195:x
1188:)
1184:y
1177:=
1174:x
1171:(
1161:)
1158:y
1155:=
1151:x
1144:(
1137:x
1134:=
1126:)
1122:x
1115:(
1104:0
1101:=
1093:1
1082:1
1079:=
1071:0
1030:y
1023:x
1016:=
1013:y
1007:x
997:y
993:x
956:(
803::
786::
767::
750:)
741::
722::
703::
684::
665::
641::
622::
582:.
493:.
404:.
303::
229:5
225:4
221:3
217:2
213:1
206::
190:·
184:·
176:·
169:·
163:·
157:·
151:·
146:(
58:.
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