Set theory question regarding $Aoverlinesim B = x+y : in Atimes B$ The Next CEO of Stack OverflowAntisymmetric's Opposite (If existant)Am I correct? State the necessary and sufficient condition for R to be an equivalence relation on A.Determining the properties for the relation over $P(mathbbN)$ where $ARB iff A cup B in H$Properties of the relation $R=(x,y)inBbb R^2$I need help proving that a relationship is not anti-symmetricValue assignment for complete, transitive relationShowing $R$ is transitive and reflexive $to$ $R=R^2$, $R$ is transitive and reflexive $to$ $R=R^2$Is the relation A = (a,a), (c,c), (d,d), (b,a) transitive?Is $asim b$ exactly when $a times b$ is divisible by $3$ an equivalence relation?A relation $R$ is defined on $mathbbZ$ by $aRb$ if and only if $2a + 2bequiv 0pmod 4$. Prove that $R$ is an equivalence relation.
Ising model simulation
Calculating discount not working
What is the difference between 'contrib' and 'non-free' packages repositories?
Is it a bad idea to plug the other end of ESD strap to wall ground?
Oldie but Goldie
What difference does it make matching a word with/without a trailing whitespace?
Is this a new Fibonacci Identity?
Is it correct to say moon starry nights?
What steps are necessary to read a Modern SSD in Medieval Europe?
How can a day be of 24 hours?
pgfplots: How to draw a tangent graph below two others?
How to find if SQL server backup is encrypted with TDE without restoring the backup
Why can't we say "I have been having a dog"?
How seriously should I take size and weight limits of hand luggage?
Why was Sir Cadogan fired?
Prodigo = pro + ago?
Calculate the Mean mean of two numbers
Would a grinding machine be a simple and workable propulsion system for an interplanetary spacecraft?
Can this transistor (2N2222) take 6 V on emitter-base? Am I reading the datasheet incorrectly?
Direct Implications Between USA and UK in Event of No-Deal Brexit
Car headlights in a world without electricity
How to implement Comparable so it is consistent with identity-equality
A hang glider, sudden unexpected lift to 25,000 feet altitude, what could do this?
Is a distribution that is normal, but highly skewed, considered Gaussian?
Set theory question regarding $Aoverlinesim B = x+y : in Atimes B$
The Next CEO of Stack OverflowAntisymmetric's Opposite (If existant)Am I correct? State the necessary and sufficient condition for R to be an equivalence relation on A.Determining the properties for the relation over $P(mathbbN)$ where $ARB iff A cup B in H$Properties of the relation $R=x-yin Bbb Z$I need help proving that a relationship is not anti-symmetricValue assignment for complete, transitive relationShowing $R$ is transitive and reflexive $to$ $R=R^2$, $R$ is transitive and reflexive $to$ $R=R^2$Is the relation A = (a,a), (c,c), (d,d), (b,a) transitive?Is $asim b$ exactly when $a times b$ is divisible by $3$ an equivalence relation?A relation $R$ is defined on $mathbbZ$ by $aRb$ if and only if $2a + 2bequiv 0pmod 4$. Prove that $R$ is an equivalence relation.
$begingroup$
Given $A,B in P(N)$ We mark $overlinesim$ as
$$Aoverlinesim B = x+y : langle x,yranglein Atimes B$$
Now, order R will be as following
$ARB$ iff $exists Min P(N)$ so $Aoverlinesim M=B$.
The question is
R is reflexive? symmetric? anti-symmetric? transitive?
I think R is partial order, i.e reflexive, anti-symmetric and transitive.
and I need to prove it.
I think my proof is correct for reflexive so let`s focus on anti-symmetric and transitive.
Just for make it sure - N is Set of Natural numbers, including zero.
Here what I done for proving transitive:
Let $A,B,C in P(N)$ and assume ARB and BRC.
ARB so exists $T in P(N)$ so $Aoverlinesim T=B$
BRC so exists $S in P(N)$ so $Boverlinesim S=C$
Let define M as following set: $K+P $
$S,Tin P(N)$ so $Min P(N)$ we need to show $Aoverlinesim M=C$
Let $qin Aoverlinesim M$
here I got stuck, since I have no idea how to show that $qin C$.
Might this R is not transitive?
Anti-Symmetric:
Let $A,Bin P(N)$ and assume ARB and BRA we need to show A=B.
let $ain A$
ARB so exists $Min P(N)$ so $Aoverlinesim M=B$
and here I got stuck.
Can we say that M is a set with just a zero element?
we have exists as not as target so we can`t choose it, I think..
Might R is not anti-symmetric?
Any help would be appreciated.
proof-verification proof-writing relations
asked Mar 24 at 18:46
John DJohn D
727
$endgroup$
add a comment |
$begingroup$
Given $A,B in P(N)$ We mark $overlinesim$ as
$$Aoverlinesim B = x+y : langle x,yranglein Atimes B$$
Now, order R will be as following
$ARB$ iff $exists Min P(N)$ so $Aoverlinesim M=B$.
The question is
R is reflexive? symmetric? anti-symmetric? transitive?
I think R is partial order, i.e reflexive, anti-symmetric and transitive.
and I need to prove it.
I think my proof is correct for reflexive so let`s focus on anti-symmetric and transitive.
Just for make it sure - N is Set of Natural numbers, including zero.
Here what I done for proving transitive:
Let $A,B,C in P(N)$ and assume ARB and BRC.
ARB so exists $T in P(N)$ so $Aoverlinesim T=B$
BRC so exists $S in P(N)$ so $Boverlinesim S=C$
Let define M as following set: $K+P $
$S,Tin P(N)$ so $Min P(N)$ we need to show $Aoverlinesim M=C$
Let $qin Aoverlinesim M$
here I got stuck, since I have no idea how to show that $qin C$.
Might this R is not transitive?
Anti-Symmetric:
Let $A,Bin P(N)$ and assume ARB and BRA we need to show A=B.
let $ain A$
ARB so exists $Min P(N)$ so $Aoverlinesim M=B$
and here I got stuck.
Can we say that M is a set with just a zero element?
we have exists as not as target so we can`t choose it, I think..
Might R is not anti-symmetric?
Any help would be appreciated.
proof-verification proof-writing relations
asked Mar 24 at 18:46
John DJohn D
727
$endgroup$
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
2
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get vialangle ranglein LaTeX.
$endgroup$
– user21820
Mar 25 at 13:52
add a comment |
$begingroup$
Given $A,B in P(N)$ We mark $overlinesim$ as
$$Aoverlinesim B = x+y : langle x,yranglein Atimes B$$
Now, order R will be as following
$ARB$ iff $exists Min P(N)$ so $Aoverlinesim M=B$.
The question is
R is reflexive? symmetric? anti-symmetric? transitive?
I think R is partial order, i.e reflexive, anti-symmetric and transitive.
and I need to prove it.
I think my proof is correct for reflexive so let`s focus on anti-symmetric and transitive.
Just for make it sure - N is Set of Natural numbers, including zero.
Here what I done for proving transitive:
Let $A,B,C in P(N)$ and assume ARB and BRC.
ARB so exists $T in P(N)$ so $Aoverlinesim T=B$
BRC so exists $S in P(N)$ so $Boverlinesim S=C$
Let define M as following set: $K+P $
$S,Tin P(N)$ so $Min P(N)$ we need to show $Aoverlinesim M=C$
Let $qin Aoverlinesim M$
here I got stuck, since I have no idea how to show that $qin C$.
Might this R is not transitive?
Anti-Symmetric:
Let $A,Bin P(N)$ and assume ARB and BRA we need to show A=B.
let $ain A$
ARB so exists $Min P(N)$ so $Aoverlinesim M=B$
and here I got stuck.
Can we say that M is a set with just a zero element?
we have exists as not as target so we can`t choose it, I think..
Might R is not anti-symmetric?
Any help would be appreciated.
proof-verification proof-writing relations
asked Mar 24 at 18:46
John DJohn D
727
$endgroup$
Given $A,B in P(N)$ We mark $overlinesim$ as
$$Aoverlinesim B = x+y : langle x,yranglein Atimes B$$
Now, order R will be as following
$ARB$ iff $exists Min P(N)$ so $Aoverlinesim M=B$.
The question is
R is reflexive? symmetric? anti-symmetric? transitive?
I think R is partial order, i.e reflexive, anti-symmetric and transitive.
and I need to prove it.
I think my proof is correct for reflexive so let`s focus on anti-symmetric and transitive.
Just for make it sure - N is Set of Natural numbers, including zero.
Here what I done for proving transitive:
Let $A,B,C in P(N)$ and assume ARB and BRC.
ARB so exists $T in P(N)$ so $Aoverlinesim T=B$
BRC so exists $S in P(N)$ so $Boverlinesim S=C$
Let define M as following set: $K+P $
$S,Tin P(N)$ so $Min P(N)$ we need to show $Aoverlinesim M=C$
Let $qin Aoverlinesim M$
here I got stuck, since I have no idea how to show that $qin C$.
Might this R is not transitive?
Anti-Symmetric:
Let $A,Bin P(N)$ and assume ARB and BRA we need to show A=B.
let $ain A$
ARB so exists $Min P(N)$ so $Aoverlinesim M=B$
and here I got stuck.
Can we say that M is a set with just a zero element?
we have exists as not as target so we can`t choose it, I think..
Might R is not anti-symmetric?
Any help would be appreciated.
proof-verification proof-writing relations
proof-verification proof-writing relations
asked Mar 24 at 18:46
John DJohn D
727
asked Mar 24 at 18:46
John DJohn D
727
edited Mar 25 at 13:54
John D
asked Mar 24 at 18:46
John DJohn D
727
asked Mar 24 at 18:46
John DJohn D
727
asked Mar 24 at 18:46
John DJohn D
727
727
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
2
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get vialangle ranglein LaTeX.
$endgroup$
– user21820
Mar 25 at 13:52
add a comment |
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
2
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get vialangle ranglein LaTeX.
$endgroup$
– user21820
Mar 25 at 13:52
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
2
2
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get via
langle rangle in LaTeX.$endgroup$
– user21820
Mar 25 at 13:52
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get via
langle rangle in LaTeX.$endgroup$
– user21820
Mar 25 at 13:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The usual notation for $overlinesim$ is +.
Show + is associative and communitive.
Clearly, A + 0 = A, so ARA.
Assume ARB, BRC. Thus exists K,L with A + K = B, B + L = C.
As A + (K + L) = (A + K) + L = B + L = C, ARC.
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
$endgroup$
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
add a comment |
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
);
);
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%2f3160866%2fset-theory-question-regarding-a-overline-sim-b-xy-x-y-in-a-times-b%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The usual notation for $overlinesim$ is +.
Show + is associative and communitive.
Clearly, A + 0 = A, so ARA.
Assume ARB, BRC. Thus exists K,L with A + K = B, B + L = C.
As A + (K + L) = (A + K) + L = B + L = C, ARC.
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
$endgroup$
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
add a comment |
$begingroup$
The usual notation for $overlinesim$ is +.
Show + is associative and communitive.
Clearly, A + 0 = A, so ARA.
Assume ARB, BRC. Thus exists K,L with A + K = B, B + L = C.
As A + (K + L) = (A + K) + L = B + L = C, ARC.
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
$endgroup$
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
add a comment |
$begingroup$
The usual notation for $overlinesim$ is +.
Show + is associative and communitive.
Clearly, A + 0 = A, so ARA.
Assume ARB, BRC. Thus exists K,L with A + K = B, B + L = C.
As A + (K + L) = (A + K) + L = B + L = C, ARC.
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
$endgroup$
The usual notation for $overlinesim$ is +.
Show + is associative and communitive.
Clearly, A + 0 = A, so ARA.
Assume ARB, BRC. Thus exists K,L with A + K = B, B + L = C.
As A + (K + L) = (A + K) + L = B + L = C, ARC.
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
edited Mar 28 at 10:46
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
answered Mar 25 at 1:11
William ElliotWilliam Elliot
8,9212820
8,9212820
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
add a comment |
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
Eilot In my proof, I can't use assoiative so it should be clear without using shortcuts like this. Can you show how to prove it stright forward?
$endgroup$
– John D
Mar 25 at 11:04
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@JohnD: If you can't assume it, prove it. It's not a good idea to say "I can't" without trying first.
$endgroup$
– user21820
Mar 25 at 13:46
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@user21820 I have started to prove it as you can see in the original post. you can see my proof. however, I got stuck
$endgroup$
– John D
Mar 25 at 13:47
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@JohnD: You misread my comment. You said you "can't use assoiative" (spelling error yours). I said "prove it" (so that you can use it).
$endgroup$
– user21820
Mar 25 at 13:49
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
$begingroup$
@William I think, transitive I have done. Can you give me hint for proving anti-symmetric?
$endgroup$
– John D
Mar 26 at 11:52
add a comment |
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.
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%2f3160866%2fset-theory-question-regarding-a-overline-sim-b-xy-x-y-in-a-times-b%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
$begingroup$
<x,y> is an ordered pair
$endgroup$
– John D
Mar 25 at 13:48
2
$begingroup$
By the way, unrelated to your question, "<" and ">" are comparison operators. The correct symbols for tuples are "⟨ ... ⟩", which you can either copy-paste as unicode symbols or get via
langle ranglein LaTeX.$endgroup$
– user21820
Mar 25 at 13:52