Mixing
If a set can't contain identical elements, how can one element be less than or EQUAL to another element
0
$begingroup$
I'm studying set theory. I understand that a set can't contain identical elements. However when I read about partially ordered sets and the introduction of "$leq$" I get confused by some sentences like:
"Given elements $a,b$ in a set $L$ we impose the axiom:
If $aleq b$ and $bleq a$, then $a=b$."
How can $a=b$ if the set can't contain identical elements?
elementary-set-theory
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
$endgroup$
add a comment |
0
$begingroup$
I'm studying set theory. I understand that a set can't contain identical elements. However when I read about partially ordered sets and the introduction of "$leq$" I get confused by some sentences like:
"Given elements $a,b$ in a set $L$ we impose the axiom:
If $aleq b$ and $bleq a$, then $a=b$."
How can $a=b$ if the set can't contain identical elements?
elementary-set-theory
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
$endgroup$
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
4
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11
add a comment |
0
0
0
$begingroup$
I'm studying set theory. I understand that a set can't contain identical elements. However when I read about partially ordered sets and the introduction of "$leq$" I get confused by some sentences like:
"Given elements $a,b$ in a set $L$ we impose the axiom:
If $aleq b$ and $bleq a$, then $a=b$."
How can $a=b$ if the set can't contain identical elements?
elementary-set-theory
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
$endgroup$
I'm studying set theory. I understand that a set can't contain identical elements. However when I read about partially ordered sets and the introduction of "$leq$" I get confused by some sentences like:
"Given elements $a,b$ in a set $L$ we impose the axiom:
If $aleq b$ and $bleq a$, then $a=b$."
How can $a=b$ if the set can't contain identical elements?
elementary-set-theory
elementary-set-theory
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
edited Jan 24 at 19:12
Carl Mummert
67.5k7133251
67.5k7133251
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
asked Jan 24 at 19:06
HeuristicsHeuristics
536212
536212
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
4
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11
add a comment |
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
4
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
4
4
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11
add a comment |
1 Answer
1
active
oldest
votes
0
$begingroup$
Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.
answered Jan 24 at 19:21
Jagol95Jagol95
1637
$endgroup$
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
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%2f3086247%2fif-a-set-cant-contain-identical-elements-how-can-one-element-be-less-than-or-e%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
0
$begingroup$
Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.
answered Jan 24 at 19:21
Jagol95Jagol95
1637
$endgroup$
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
add a comment |
0
$begingroup$
Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.
answered Jan 24 at 19:21
Jagol95Jagol95
1637
$endgroup$
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
add a comment |
0
0
0
$begingroup$
Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.
answered Jan 24 at 19:21
Jagol95Jagol95
1637
$endgroup$
Here is an analogy. No two people are the same. If Person A has the same mother as Person B and they are not twins and where born on the same day then Person A and Person B are the same Person.
answered Jan 24 at 19:21
Jagol95Jagol95
1637
answered Jan 24 at 19:21
Jagol95Jagol95
1637
answered Jan 24 at 19:21
Jagol95Jagol95
1637
answered Jan 24 at 19:21
Jagol95Jagol95
1637
1637
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
add a comment |
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
"No two people people are the same" No, no two distinct people are the same. I am a person, and I am also a person. We are the same person. More generally, different variables need not refer to different elements; if $a,bin P$ then it might be the case that $a = b$.
$endgroup$
– Mario Carneiro
Jan 24 at 20:00
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
It is not a formal example so I think your comment raises more questions than it answers. Wouldn't the statement that no two distinct people are the same be redundant? If there were two seperate people who where identical in every way would they be considered distinct? Clearly trying to answer such questions would purely be a matter of definitions, of which there are none at hand for this example.
$endgroup$
– Jagol95
Jan 24 at 20:15
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
$begingroup$
Yes, the statement that no two distinct people are the same is tautologous. It amounts to the statement $forall x y,xne yto xne y$. But it is important to recognize the difference between this (trivially true) statement and the false statement $forall x y,xne y$. In English, we often insert an implicit distinctness condition on double quantifiers (i.e. "two people cannot be in the same place at the same time"), but mathematical english does not do this (when used carefully), and it is important to notice this.
$endgroup$
– Mario Carneiro
Jan 25 at 1:52
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%2f3086247%2fif-a-set-cant-contain-identical-elements-how-can-one-element-be-less-than-or-e%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
$begingroup$
"Can't contain identical elements" should be read as "Does not contain multiple copies of any elements"
$endgroup$
– Morgan Rodgers
Jan 24 at 19:08
$begingroup$
It's misleading to say that a set cannot contain identical elements. It's just that if an element is repeated, then the set is exactly the same.
$endgroup$
– Wojowu
Jan 24 at 19:08
4
$begingroup$
Different variables can still represent the same element of a set.
$endgroup$
– Carl Mummert
Jan 24 at 19:11