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
asked Nov 27 '18 at 21:01
Arthur Green
776
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
asked Nov 27 '18 at 21:01
Arthur Green
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 |
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
asked Nov 27 '18 at 21:01
Arthur Green
776
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
asked Nov 27 '18 at 21:01
Arthur Green
776
asked Nov 27 '18 at 21:01
Arthur Green
776
edited Nov 27 '18 at 21:19
asked Nov 27 '18 at 21:01
Arthur Green
776
asked Nov 27 '18 at 21:01
Arthur Green
776
asked Nov 27 '18 at 21:01
Arthur 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 |
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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