Mixing
If ∩F and ∩G are disjoint, then for some A ∈ F and B ∈ G, A and B are disjoint.
If ∩F and ∩G are disjoint, then for some A ∈ F and B ∈ G, A and B are disjoint.
Give a proof or counter-example.
I'm like 99 percent the theorem is true, can't seem to prove it. Have tried a few methods, closest I have got is trying to prove the contra-positive. Any help with this problem would be extremely appreciated, thanks in advance!
elementary-set-theory proof-writing
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
|
show 1 more comment
If ∩F and ∩G are disjoint, then for some A ∈ F and B ∈ G, A and B are disjoint.
Give a proof or counter-example.
I'm like 99 percent the theorem is true, can't seem to prove it. Have tried a few methods, closest I have got is trying to prove the contra-positive. Any help with this problem would be extremely appreciated, thanks in advance!
elementary-set-theory proof-writing
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
3
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
1
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
1
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
1
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25
|
show 1 more comment
If ∩F and ∩G are disjoint, then for some A ∈ F and B ∈ G, A and B are disjoint.
Give a proof or counter-example.
I'm like 99 percent the theorem is true, can't seem to prove it. Have tried a few methods, closest I have got is trying to prove the contra-positive. Any help with this problem would be extremely appreciated, thanks in advance!
elementary-set-theory proof-writing
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
If ∩F and ∩G are disjoint, then for some A ∈ F and B ∈ G, A and B are disjoint.
Give a proof or counter-example.
I'm like 99 percent the theorem is true, can't seem to prove it. Have tried a few methods, closest I have got is trying to prove the contra-positive. Any help with this problem would be extremely appreciated, thanks in advance!
elementary-set-theory proof-writing
elementary-set-theory proof-writing
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
edited Nov 21 '18 at 7:39
астон вілла олоф мэллбэрг
37.3k33376
37.3k33376
asked Nov 21 '18 at 7:07
Anon96
1
asked Nov 21 '18 at 7:07
Anon96
1
asked Nov 21 '18 at 7:07
Anon96
1
1
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
3
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
1
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
1
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
1
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25
|
show 1 more comment
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
3
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
1
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
1
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
1
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
3
3
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
1
1
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
1
1
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
1
1
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25
|
show 1 more comment
2 Answers
2
active
oldest
votes
Counterexample : $A = {1,2,3}$, $C = {3,4,5}$, $B = {3,6,7}$, $D = {1,5,6}$ with $F = {A,C}$ and $G = {B,D}$.
In particular, $3 in A cap B$, $3 in C cap B$, $5 in C cap D$ , $1 in A cap D$. However, $cap F = {3}$ and $cap G = {6}$ are disjoint.
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
add a comment |
Think small: $mathcal{F}=bigl{{1,2}bigr}$, $mathcal{G}=bigl{{1},{2}bigr}$.
If you don't like that the sets have empty intersection, use $mathcal{G}=bigl{{1,3},{2,3}bigr}$
answered Nov 21 '18 at 8:57
egreg
178k1484201
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%2f3007366%2fif-%25e2%2588%25a9f-and-%25e2%2588%25a9g-are-disjoint-then-for-some-a-%25e2%2588%2588-f-and-b-%25e2%2588%2588-g-a-and-b-are-disjoint%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
Counterexample : $A = {1,2,3}$, $C = {3,4,5}$, $B = {3,6,7}$, $D = {1,5,6}$ with $F = {A,C}$ and $G = {B,D}$.
In particular, $3 in A cap B$, $3 in C cap B$, $5 in C cap D$ , $1 in A cap D$. However, $cap F = {3}$ and $cap G = {6}$ are disjoint.
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
add a comment |
Counterexample : $A = {1,2,3}$, $C = {3,4,5}$, $B = {3,6,7}$, $D = {1,5,6}$ with $F = {A,C}$ and $G = {B,D}$.
In particular, $3 in A cap B$, $3 in C cap B$, $5 in C cap D$ , $1 in A cap D$. However, $cap F = {3}$ and $cap G = {6}$ are disjoint.
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
add a comment |
Counterexample : $A = {1,2,3}$, $C = {3,4,5}$, $B = {3,6,7}$, $D = {1,5,6}$ with $F = {A,C}$ and $G = {B,D}$.
In particular, $3 in A cap B$, $3 in C cap B$, $5 in C cap D$ , $1 in A cap D$. However, $cap F = {3}$ and $cap G = {6}$ are disjoint.
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
Counterexample : $A = {1,2,3}$, $C = {3,4,5}$, $B = {3,6,7}$, $D = {1,5,6}$ with $F = {A,C}$ and $G = {B,D}$.
In particular, $3 in A cap B$, $3 in C cap B$, $5 in C cap D$ , $1 in A cap D$. However, $cap F = {3}$ and $cap G = {6}$ are disjoint.
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
answered Nov 21 '18 at 8:25
астон вілла олоф мэллбэрг
37.3k33376
37.3k33376
add a comment |
add a comment |
Think small: $mathcal{F}=bigl{{1,2}bigr}$, $mathcal{G}=bigl{{1},{2}bigr}$.
If you don't like that the sets have empty intersection, use $mathcal{G}=bigl{{1,3},{2,3}bigr}$
answered Nov 21 '18 at 8:57
egreg
178k1484201
add a comment |
Think small: $mathcal{F}=bigl{{1,2}bigr}$, $mathcal{G}=bigl{{1},{2}bigr}$.
If you don't like that the sets have empty intersection, use $mathcal{G}=bigl{{1,3},{2,3}bigr}$
answered Nov 21 '18 at 8:57
egreg
178k1484201
add a comment |
Think small: $mathcal{F}=bigl{{1,2}bigr}$, $mathcal{G}=bigl{{1},{2}bigr}$.
If you don't like that the sets have empty intersection, use $mathcal{G}=bigl{{1,3},{2,3}bigr}$
answered Nov 21 '18 at 8:57
egreg
178k1484201
Think small: $mathcal{F}=bigl{{1,2}bigr}$, $mathcal{G}=bigl{{1},{2}bigr}$.
If you don't like that the sets have empty intersection, use $mathcal{G}=bigl{{1,3},{2,3}bigr}$
answered Nov 21 '18 at 8:57
egreg
178k1484201
answered Nov 21 '18 at 8:57
egreg
178k1484201
answered Nov 21 '18 at 8:57
egreg
178k1484201
answered Nov 21 '18 at 8:57
egreg
178k1484201
178k1484201
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3007366%2fif-%25e2%2588%25a9f-and-%25e2%2588%25a9g-are-disjoint-then-for-some-a-%25e2%2588%2588-f-and-b-%25e2%2588%2588-g-a-and-b-are-disjoint%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
Have you looked for a counterexample?
– Lord Shark the Unknown
Nov 21 '18 at 7:09
3
The above counterexample does not work, since $A cap D$ and $C cap D$ are both empty. However, by taking $D = {1,6,5}$ I think we are done : although $A cap C = {3}$ and $B cap D = {6}$ are disjoint, we see that $3 in A cap B, 1 in A cap D, 3 in C cap B$ and $5 in C cap D$ are true. So $F = {A,C}$ and $G = {B,D}$ serves as a counterexample.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 7:22
1
@астонвіллаолофмэллбэрг, your comment qualifies as an actual answer to the OP. Please post it as an actual answer, so that the question does not remain unanswered.
– Jose Arnaldo Bebita Dris
Nov 21 '18 at 8:08
1
karagila.org/2015/how-to-solve-your-problems
– Asaf Karagila♦
Nov 21 '18 at 8:16
1
@JoseArnaldoBebitaDris Thank you, I have done it below.
– астон вілла олоф мэллбэрг
Nov 21 '18 at 8:25