Mixing
Construct algebras $A_p subseteq mathscr(P) (X)$ with cardinality $|A_p | = 2^p$.
$begingroup$
If $X$ is a set with $|X| geq s$, for every $p in {1,2,...,s} $, construct an algebra $mathscr{A} _p subseteq mathscr{P} (X)$ such that $|mathscr{A} _p | = 2^p$.
I was thinking on doing this by induction on $p$. For $p=1$, we can take $mathscr{A} _1 = {X, emptyset }$. Now, if $mathscr{A} _{p-1}$ is an algebra with $2^{p-1}$ elements, then I was thinking that I could take $Bsubseteq X$ not contained in $mathscr{A} _{p-1}$, and then consider $$mathscr{A} _p = mathscr{A} _{p-1} cup { Bcup A | A in mathscr{A} _{p-1} } cup { B^c cap A | A in mathscr{A} _{p-1} } ,$$ but I'm having trouble showing that this set is closed under finite unions. I also don't know if this set has cardinality $2^p$.
I would be very thankful if you could help me with this!
real-analysis probability measure-theory boolean-algebra boolean-ring
asked Jan 30 at 4:58
user392559user392559
38118
$endgroup$
add a comment |
$begingroup$
If $X$ is a set with $|X| geq s$, for every $p in {1,2,...,s} $, construct an algebra $mathscr{A} _p subseteq mathscr{P} (X)$ such that $|mathscr{A} _p | = 2^p$.
I was thinking on doing this by induction on $p$. For $p=1$, we can take $mathscr{A} _1 = {X, emptyset }$. Now, if $mathscr{A} _{p-1}$ is an algebra with $2^{p-1}$ elements, then I was thinking that I could take $Bsubseteq X$ not contained in $mathscr{A} _{p-1}$, and then consider $$mathscr{A} _p = mathscr{A} _{p-1} cup { Bcup A | A in mathscr{A} _{p-1} } cup { B^c cap A | A in mathscr{A} _{p-1} } ,$$ but I'm having trouble showing that this set is closed under finite unions. I also don't know if this set has cardinality $2^p$.
I would be very thankful if you could help me with this!
real-analysis probability measure-theory boolean-algebra boolean-ring
asked Jan 30 at 4:58
user392559user392559
38118
$endgroup$
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13
add a comment |
$begingroup$
If $X$ is a set with $|X| geq s$, for every $p in {1,2,...,s} $, construct an algebra $mathscr{A} _p subseteq mathscr{P} (X)$ such that $|mathscr{A} _p | = 2^p$.
I was thinking on doing this by induction on $p$. For $p=1$, we can take $mathscr{A} _1 = {X, emptyset }$. Now, if $mathscr{A} _{p-1}$ is an algebra with $2^{p-1}$ elements, then I was thinking that I could take $Bsubseteq X$ not contained in $mathscr{A} _{p-1}$, and then consider $$mathscr{A} _p = mathscr{A} _{p-1} cup { Bcup A | A in mathscr{A} _{p-1} } cup { B^c cap A | A in mathscr{A} _{p-1} } ,$$ but I'm having trouble showing that this set is closed under finite unions. I also don't know if this set has cardinality $2^p$.
I would be very thankful if you could help me with this!
real-analysis probability measure-theory boolean-algebra boolean-ring
asked Jan 30 at 4:58
user392559user392559
38118
$endgroup$
If $X$ is a set with $|X| geq s$, for every $p in {1,2,...,s} $, construct an algebra $mathscr{A} _p subseteq mathscr{P} (X)$ such that $|mathscr{A} _p | = 2^p$.
I was thinking on doing this by induction on $p$. For $p=1$, we can take $mathscr{A} _1 = {X, emptyset }$. Now, if $mathscr{A} _{p-1}$ is an algebra with $2^{p-1}$ elements, then I was thinking that I could take $Bsubseteq X$ not contained in $mathscr{A} _{p-1}$, and then consider $$mathscr{A} _p = mathscr{A} _{p-1} cup { Bcup A | A in mathscr{A} _{p-1} } cup { B^c cap A | A in mathscr{A} _{p-1} } ,$$ but I'm having trouble showing that this set is closed under finite unions. I also don't know if this set has cardinality $2^p$.
I would be very thankful if you could help me with this!
real-analysis probability measure-theory boolean-algebra boolean-ring
real-analysis probability measure-theory boolean-algebra boolean-ring
asked Jan 30 at 4:58
user392559user392559
38118
asked Jan 30 at 4:58
user392559user392559
38118
edited Jan 30 at 5:51
user635162
asked Jan 30 at 4:58
user392559user392559
38118
asked Jan 30 at 4:58
user392559user392559
38118
asked Jan 30 at 4:58
user392559user392559
38118
38118
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13
add a comment |
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13
add a comment |
0
active
oldest
votes
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%2f3093115%2fconstruct-algebras-a-p-subseteq-mathscrp-x-with-cardinality-a-p-2%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f3093115%2fconstruct-algebras-a-p-subseteq-mathscrp-x-with-cardinality-a-p-2%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
$begingroup$
Note that in latex an uppercase Alpha is just "A", not "Alpha"
$endgroup$
– user635162
Jan 30 at 5:13