equivalence relation with binary and decimal numbers
I am learning about relations and I was hoping to find out if my attempt for my question looks right.
Let b(n) equal the value of the highest bit set to 1 in the binary representation of the positive integer n. (For example, b(27)=16 because in 27= $11011_2$ and the most significant bit set to one is the first bit on the left, which has value $2^4$.)
Prove that the relation, R, defined below over the set of integers in between 0 and 1023, inclusive, is an equivalence relation. Into how many equivalence classes does R partition the set described? Explicitly list all of the members of the following equivalence classes: [2] and [13]. Let the set X be the largest of the equivalence classes. What is the smallest integer that belongs to X?
$$R = { (x,y) | b(x) = b(y) } $$
- R is reflexive as $b(i) = b(i) $ for all $i in Z$.
- R is symmetric as $b(i) = b(j) rightarrow b(j) = b(i)$
- R is transitive as $b(i) = b(j) wedge B(j) = b(k)$ then $b(i) = b(k)$
[2] = $ {2, 2 } $ (2 in binary is 10)
[13] = $ {13, 4 } $ (13 in binary is 1101)
Would 512 make sense to be the smallest integer?
equivalence-relations
add a comment |
I am learning about relations and I was hoping to find out if my attempt for my question looks right.
Let b(n) equal the value of the highest bit set to 1 in the binary representation of the positive integer n. (For example, b(27)=16 because in 27= $11011_2$ and the most significant bit set to one is the first bit on the left, which has value $2^4$.)
Prove that the relation, R, defined below over the set of integers in between 0 and 1023, inclusive, is an equivalence relation. Into how many equivalence classes does R partition the set described? Explicitly list all of the members of the following equivalence classes: [2] and [13]. Let the set X be the largest of the equivalence classes. What is the smallest integer that belongs to X?
$$R = { (x,y) | b(x) = b(y) } $$
- R is reflexive as $b(i) = b(i) $ for all $i in Z$.
- R is symmetric as $b(i) = b(j) rightarrow b(j) = b(i)$
- R is transitive as $b(i) = b(j) wedge B(j) = b(k)$ then $b(i) = b(k)$
[2] = $ {2, 2 } $ (2 in binary is 10)
[13] = $ {13, 4 } $ (13 in binary is 1101)
Would 512 make sense to be the smallest integer?
equivalence-relations
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
2
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12
add a comment |
I am learning about relations and I was hoping to find out if my attempt for my question looks right.
Let b(n) equal the value of the highest bit set to 1 in the binary representation of the positive integer n. (For example, b(27)=16 because in 27= $11011_2$ and the most significant bit set to one is the first bit on the left, which has value $2^4$.)
Prove that the relation, R, defined below over the set of integers in between 0 and 1023, inclusive, is an equivalence relation. Into how many equivalence classes does R partition the set described? Explicitly list all of the members of the following equivalence classes: [2] and [13]. Let the set X be the largest of the equivalence classes. What is the smallest integer that belongs to X?
$$R = { (x,y) | b(x) = b(y) } $$
- R is reflexive as $b(i) = b(i) $ for all $i in Z$.
- R is symmetric as $b(i) = b(j) rightarrow b(j) = b(i)$
- R is transitive as $b(i) = b(j) wedge B(j) = b(k)$ then $b(i) = b(k)$
[2] = $ {2, 2 } $ (2 in binary is 10)
[13] = $ {13, 4 } $ (13 in binary is 1101)
Would 512 make sense to be the smallest integer?
equivalence-relations
I am learning about relations and I was hoping to find out if my attempt for my question looks right.
Let b(n) equal the value of the highest bit set to 1 in the binary representation of the positive integer n. (For example, b(27)=16 because in 27= $11011_2$ and the most significant bit set to one is the first bit on the left, which has value $2^4$.)
Prove that the relation, R, defined below over the set of integers in between 0 and 1023, inclusive, is an equivalence relation. Into how many equivalence classes does R partition the set described? Explicitly list all of the members of the following equivalence classes: [2] and [13]. Let the set X be the largest of the equivalence classes. What is the smallest integer that belongs to X?
$$R = { (x,y) | b(x) = b(y) } $$
- R is reflexive as $b(i) = b(i) $ for all $i in Z$.
- R is symmetric as $b(i) = b(j) rightarrow b(j) = b(i)$
- R is transitive as $b(i) = b(j) wedge B(j) = b(k)$ then $b(i) = b(k)$
[2] = $ {2, 2 } $ (2 in binary is 10)
[13] = $ {13, 4 } $ (13 in binary is 1101)
Would 512 make sense to be the smallest integer?
equivalence-relations
equivalence-relations
edited Nov 27 '18 at 21:19
Arthur Green
asked Nov 27 '18 at 21:01
Arthur GreenArthur Green
776
776
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
2
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12
add a comment |
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
2
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
2
2
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12
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%2f3016291%2fequivalence-relation-with-binary-and-decimal-numbers%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.
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%2f3016291%2fequivalence-relation-with-binary-and-decimal-numbers%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
Your equivalence classes are incorrect. For example, $[2]={2,3}$ and $[13]={8,9,10,11,12,13,14,15}$ and
– Anurag A
Nov 27 '18 at 21:06
@AnuragA the largest 1 in binary form of 2 is (10). How would it be three?
– Arthur Green
Nov 27 '18 at 21:07
2
The binary representation of $3=(11)_2$. So $b(3)=2$. Thus $2$ and $3$ are equivalent under this relation. Another way to understand this relation is as follows: two integers are related if the highest power of $2$ that is less than or equal to those integers is same.
– Anurag A
Nov 27 '18 at 21:12