Is it possible to express disjunction through conjunction and implication?
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Is it possible to express disjunction through conjunction and implication?
$begingroup$
This question is about Boolean functions. Is it possible to express disjunction $xlor y$ through conjunction $xland y$ (or simply $xy$) and implication $xto y$?
boolean-algebra
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
$endgroup$
add a comment |
$begingroup$
This question is about Boolean functions. Is it possible to express disjunction $xlor y$ through conjunction $xland y$ (or simply $xy$) and implication $xto y$?
boolean-algebra
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
$endgroup$
$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
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@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago
add a comment |
$begingroup$
This question is about Boolean functions. Is it possible to express disjunction $xlor y$ through conjunction $xland y$ (or simply $xy$) and implication $xto y$?
boolean-algebra
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
$endgroup$
This question is about Boolean functions. Is it possible to express disjunction $xlor y$ through conjunction $xland y$ (or simply $xy$) and implication $xto y$?
boolean-algebra
boolean-algebra
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
asked 9 hours ago
Evgeny MakarovEvgeny Makarov
363
363
$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
$begingroup$
@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago
add a comment |
$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
$begingroup$
@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago
$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
$begingroup$
@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Yes - in fact, disjunction is expressible via implication alone:
$$
x lor y equiv (x to y) to y
$$
answered 9 hours ago
AdrianoAdriano
36.7k33371
$endgroup$
add a comment |
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1 Answer
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active
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1 Answer
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oldest
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$begingroup$
Yes - in fact, disjunction is expressible via implication alone:
$$
x lor y equiv (x to y) to y
$$
answered 9 hours ago
AdrianoAdriano
36.7k33371
$endgroup$
add a comment |
$begingroup$
Yes - in fact, disjunction is expressible via implication alone:
$$
x lor y equiv (x to y) to y
$$
answered 9 hours ago
AdrianoAdriano
36.7k33371
$endgroup$
add a comment |
$begingroup$
Yes - in fact, disjunction is expressible via implication alone:
$$
x lor y equiv (x to y) to y
$$
answered 9 hours ago
AdrianoAdriano
36.7k33371
$endgroup$
Yes - in fact, disjunction is expressible via implication alone:
$$
x lor y equiv (x to y) to y
$$
answered 9 hours ago
AdrianoAdriano
36.7k33371
answered 9 hours ago
AdrianoAdriano
36.7k33371
answered 9 hours ago
AdrianoAdriano
36.7k33371
answered 9 hours ago
AdrianoAdriano
36.7k33371
36.7k33371
add a comment |
add a comment |
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$begingroup$
It should, since negation is expressable with implications
$endgroup$
– miraunpajaro
9 hours ago
$begingroup$
@miraunpajaro Negation expressable with implications? How?
$endgroup$
– jjagmath
8 hours ago
$begingroup$
I seem to remember implication and conjunction was functionally complete, I can't figure it out now though
$endgroup$
– miraunpajaro
7 hours ago
$begingroup$
Implication and conjunction are not functionally complete since they both map 1 (truth) into 1. Negation is expressible though implication as $xto 0$, but one needs falsehood for this. The formula in the accepted answer just slipped my mind.
$endgroup$
– Evgeny Makarov
5 hours ago