Show $p lor (p land q ) equiv p $ using equivalences
0
$begingroup$
I am trying to show $p lor (p land q ) equiv p $ using equivalences.
I have tried many replacements (e.g. distributivity and de Morgans) but cannot see a way to simplify the left hand side that reduces to $p$.
Here are is a list of logical equivalences from wikipedia.
I know that this statement is true (via truth tables), but I cannot derive this using equivalences. Ideas appreciated.
logic propositional-calculus boolean-algebra
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
$endgroup$
add a comment |
0
$begingroup$
I am trying to show $p lor (p land q ) equiv p $ using equivalences.
I have tried many replacements (e.g. distributivity and de Morgans) but cannot see a way to simplify the left hand side that reduces to $p$.
Here are is a list of logical equivalences from wikipedia.
I know that this statement is true (via truth tables), but I cannot derive this using equivalences. Ideas appreciated.
logic propositional-calculus boolean-algebra
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
$endgroup$
2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39
add a comment |
0
0
0
$begingroup$
I am trying to show $p lor (p land q ) equiv p $ using equivalences.
I have tried many replacements (e.g. distributivity and de Morgans) but cannot see a way to simplify the left hand side that reduces to $p$.
Here are is a list of logical equivalences from wikipedia.
I know that this statement is true (via truth tables), but I cannot derive this using equivalences. Ideas appreciated.
logic propositional-calculus boolean-algebra
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
$endgroup$
I am trying to show $p lor (p land q ) equiv p $ using equivalences.
I have tried many replacements (e.g. distributivity and de Morgans) but cannot see a way to simplify the left hand side that reduces to $p$.
Here are is a list of logical equivalences from wikipedia.
I know that this statement is true (via truth tables), but I cannot derive this using equivalences. Ideas appreciated.
logic propositional-calculus boolean-algebra
logic propositional-calculus boolean-algebra
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
edited Dec 14 '18 at 13:36
Bram28
60.5k44590
60.5k44590
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2241210
2241210
2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39
add a comment |
2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39
2
2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39
add a comment |
2 Answers
2
active
oldest
votes
3
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
$endgroup$
add a comment |
0
$begingroup$
It's in the list ... but some texts like to derive it as follows:
$$p lor (p land q) equiv (p land top) lor (p land q) equiv p land (top lor q) equiv p land top equiv p$$
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
$endgroup$
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
3
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
$endgroup$
add a comment |
3
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
$endgroup$
add a comment |
3
3
3
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
$endgroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
answered Dec 14 '18 at 11:41
ArthurArthur
112k7107191
112k7107191
add a comment |
add a comment |
0
$begingroup$
It's in the list ... but some texts like to derive it as follows:
$$p lor (p land q) equiv (p land top) lor (p land q) equiv p land (top lor q) equiv p land top equiv p$$
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
$endgroup$
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
add a comment |
0
$begingroup$
It's in the list ... but some texts like to derive it as follows:
$$p lor (p land q) equiv (p land top) lor (p land q) equiv p land (top lor q) equiv p land top equiv p$$
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
$endgroup$
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
add a comment |
0
0
0
$begingroup$
It's in the list ... but some texts like to derive it as follows:
$$p lor (p land q) equiv (p land top) lor (p land q) equiv p land (top lor q) equiv p land top equiv p$$
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
$endgroup$
It's in the list ... but some texts like to derive it as follows:
$$p lor (p land q) equiv (p land top) lor (p land q) equiv p land (top lor q) equiv p land top equiv p$$
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
answered Dec 14 '18 at 12:49
Bram28Bram28
60.5k44590
60.5k44590
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
add a comment |
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan That's an instance of Distribution ... which is often not derived from other equivalence principles. That is, Distribution is often a given equivalence principle.
$endgroup$
– Bram28
Dec 15 '18 at 21:00
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan By Distribution I mean $p land (q lor r) equiv (p land q) lor (p land r)$
$endgroup$
– Bram28
Dec 15 '18 at 21:53
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
@RobArthan Why not? $p=p$, $q= top$, and $r=q$ ...
$endgroup$
– Bram28
Dec 16 '18 at 0:10
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
RIght - my brain was not in gear!
$endgroup$
– Rob Arthan
Dec 16 '18 at 12:11
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
$begingroup$
@RobArthan No worries! It was probably because it went in the 'opposite' direction of what you normally understand 'Distribution' to be ... more like a 'Reverse Distribution' or 'Un-Distribution' or 'Factoring'
$endgroup$
– Bram28
Dec 16 '18 at 12:33
add a comment |
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2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39