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
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
$endgroup$
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
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
$endgroup$
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
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
$endgroup$
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
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
edited Dec 14 '18 at 13:36
Bram28
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
edited Dec 14 '18 at 13:36
Bram28
63.2k44793
edited Dec 14 '18 at 13:36
Bram28
63.2k44793
edited Dec 14 '18 at 13:36
Bram28
63.2k44793
63.2k44793
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
asked Dec 14 '18 at 11:35
Conor CosnettConor Cosnett
2281211
2281211
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
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
63.2k44793
$endgroup$
$begingroup$
@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 |
4
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039251%2fshow-p-lor-p-land-q-equiv-p-using-equivalences%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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
63.2k44793
$endgroup$
$begingroup$
@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
63.2k44793
$endgroup$
$begingroup$
@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
63.2k44793
$endgroup$
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
63.2k44793
answered Dec 14 '18 at 12:49
Bram28Bram28
63.2k44793
answered Dec 14 '18 at 12:49
Bram28Bram28
63.2k44793
answered Dec 14 '18 at 12:49
Bram28Bram28
63.2k44793
63.2k44793
$begingroup$
@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 |
4
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
$endgroup$
add a comment |
4
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
$endgroup$
add a comment |
4
4
4
$begingroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
$endgroup$
This is listed, verbatim, as one of the absorbtion laws in the wikipedia page you linked.
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
answered Dec 14 '18 at 11:41
ArthurArthur
116k7116199
116k7116199
add a comment |
add a comment |
draft saved
draft discarded
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3039251%2fshow-p-lor-p-land-q-equiv-p-using-equivalences%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
2
$begingroup$
Law of absorption. This is an axiom in lattice theory.
$endgroup$
– Wuestenfux
Dec 14 '18 at 11:39