Mixing
equivalence classes and minimal elemnts
$begingroup$
$A = {n | 2 le n le 12}$
$D$ is relation on $A$ defined: $(a,b) in D$ iff $b pmod{a} equiv 0$.
I need to find the elements which are minimal regarding to D.
What I think to do is to find all Equivalence Class and then take the minimal in each Equivalence Class. But here I think I miss something.
2 and 12 need to be in the same Equivalence Class, since 12 mod 2 === 0
2 and 10 need to be in the same Equivalence Class, since 10 mod 2 === 0
but each element can appear only in one Equivalence Class and so 2 => it mean that 2,10,12 need to be in the same class but 12 mod 10 !== 0
What I miss here?
thanks
discrete-mathematics
edited Jan 21 at 18:38
gt6989b
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
$endgroup$
add a comment |
$begingroup$
$A = {n | 2 le n le 12}$
$D$ is relation on $A$ defined: $(a,b) in D$ iff $b pmod{a} equiv 0$.
I need to find the elements which are minimal regarding to D.
What I think to do is to find all Equivalence Class and then take the minimal in each Equivalence Class. But here I think I miss something.
2 and 12 need to be in the same Equivalence Class, since 12 mod 2 === 0
2 and 10 need to be in the same Equivalence Class, since 10 mod 2 === 0
but each element can appear only in one Equivalence Class and so 2 => it mean that 2,10,12 need to be in the same class but 12 mod 10 !== 0
What I miss here?
thanks
discrete-mathematics
edited Jan 21 at 18:38
gt6989b
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
$endgroup$
2
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52
add a comment |
$begingroup$
$A = {n | 2 le n le 12}$
$D$ is relation on $A$ defined: $(a,b) in D$ iff $b pmod{a} equiv 0$.
I need to find the elements which are minimal regarding to D.
What I think to do is to find all Equivalence Class and then take the minimal in each Equivalence Class. But here I think I miss something.
2 and 12 need to be in the same Equivalence Class, since 12 mod 2 === 0
2 and 10 need to be in the same Equivalence Class, since 10 mod 2 === 0
but each element can appear only in one Equivalence Class and so 2 => it mean that 2,10,12 need to be in the same class but 12 mod 10 !== 0
What I miss here?
thanks
discrete-mathematics
edited Jan 21 at 18:38
gt6989b
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
$endgroup$
$A = {n | 2 le n le 12}$
$D$ is relation on $A$ defined: $(a,b) in D$ iff $b pmod{a} equiv 0$.
I need to find the elements which are minimal regarding to D.
What I think to do is to find all Equivalence Class and then take the minimal in each Equivalence Class. But here I think I miss something.
2 and 12 need to be in the same Equivalence Class, since 12 mod 2 === 0
2 and 10 need to be in the same Equivalence Class, since 10 mod 2 === 0
but each element can appear only in one Equivalence Class and so 2 => it mean that 2,10,12 need to be in the same class but 12 mod 10 !== 0
What I miss here?
thanks
discrete-mathematics
discrete-mathematics
edited Jan 21 at 18:38
gt6989b
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
edited Jan 21 at 18:38
gt6989b
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
edited Jan 21 at 18:38
gt6989b
34.7k22456
edited Jan 21 at 18:38
gt6989b
34.7k22456
edited Jan 21 at 18:38
gt6989b
34.7k22456
34.7k22456
asked Jan 21 at 18:36
user3523226user3523226
607
asked Jan 21 at 18:36
user3523226user3523226
607
asked Jan 21 at 18:36
user3523226user3523226
607
607
2
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52
add a comment |
2
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52
2
2
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
To say that $bequiv0pmod a$ is equivalent to saying that $amid b$. So, your relation is that $(a,b)in D$ if and only if $amid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in ${2,3,ldots,12}$ other than itself.
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
$endgroup$
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
});
}
});
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%2f3082222%2fequivalence-classes-and-minimal-elemnts%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
$begingroup$
Hint:
To say that $bequiv0pmod a$ is equivalent to saying that $amid b$. So, your relation is that $(a,b)in D$ if and only if $amid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in ${2,3,ldots,12}$ other than itself.
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
$endgroup$
add a comment |
$begingroup$
Hint:
To say that $bequiv0pmod a$ is equivalent to saying that $amid b$. So, your relation is that $(a,b)in D$ if and only if $amid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in ${2,3,ldots,12}$ other than itself.
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
$endgroup$
add a comment |
$begingroup$
Hint:
To say that $bequiv0pmod a$ is equivalent to saying that $amid b$. So, your relation is that $(a,b)in D$ if and only if $amid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in ${2,3,ldots,12}$ other than itself.
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
$endgroup$
Hint:
To say that $bequiv0pmod a$ is equivalent to saying that $amid b$. So, your relation is that $(a,b)in D$ if and only if $amid b$.
So, what sort of element is minimal in this setup? Being minimal means that a number isn't divisible by any number in ${2,3,ldots,12}$ other than itself.
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
answered Jan 23 at 23:25
Nick PetersonNick Peterson
26.8k23962
26.8k23962
add a comment |
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%2f3082222%2fequivalence-classes-and-minimal-elemnts%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
$begingroup$
This is not clear. Your relation isn't an equivalence relation...$2sim 4$ for instance, but $4not sim 2$. So...what do you mean by an equivalence class?
$endgroup$
– lulu
Jan 21 at 18:39
$begingroup$
As lulu pointed out, this is not an equivalence relation, but it is an order relation (in fact, there's a much easier way to describe this relation- recall the definition of modular congruence) so you can talk about minimal elements, i.e. elements $x$ such that if $(y,x)$, then $y=x$. If you're familiar with the idea of posets and Hasse diagrams, then draw the Hasse diagram for this poset and you'll see what the minimal element(s) are.
$endgroup$
– Kevin Long
Jan 21 at 18:43
$begingroup$
Thank for this clarification
$endgroup$
– user3523226
Jan 22 at 12:52