Prove that ${A_n}_{n in mathbb{Z}}$ is a partition of $mathbb{Z}$ where $A_n = {m in mathbb{Z}: exists q ni m...
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This is the first exercise of section 3.3 from A Book of Set Theory by Charles C. Pinter.
This is my attempt to solve it, please tell me where I'm doing wrong:
To prove that $A_n$ is a partition of $mathbb{Z}$, we must prove that:
P1: $$If:exists x in A_i cap A_j, then : A_i=A_j$$
P2: $$If :x in A, then: x in A_i :for: some: i in I$$
Suppose $A_i cap A_j$ is a non-empty set. We must prove that $A_i = A_j$
Let $y in A_i$. Then by definition of $A_n$, there exists some $k$ such that $y = i + 5k$. We should prove that $y in A_j$. Add and substract 5 from the right hand side:
$$y = i + 5k + 5 -5$$
$$y = i + 5 + 5(k - 1)$$
Now let $j = i + 1$. Then $y in A_j$ for $j = i + 5$ and $q = k + 1$.
Proof of P2:
Let $n in mathbb{Z}$. We must prove that $n in A_i$ for some $i in I$. But this can be easily shown to be true by the quotient-reminder theorem.
proof-verification elementary-set-theory proof-writing
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
asked Nov 17 at 13:36
amin kamali
11
add a comment |
up vote
0
down vote
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This is the first exercise of section 3.3 from A Book of Set Theory by Charles C. Pinter.
This is my attempt to solve it, please tell me where I'm doing wrong:
To prove that $A_n$ is a partition of $mathbb{Z}$, we must prove that:
P1: $$If:exists x in A_i cap A_j, then : A_i=A_j$$
P2: $$If :x in A, then: x in A_i :for: some: i in I$$
Suppose $A_i cap A_j$ is a non-empty set. We must prove that $A_i = A_j$
Let $y in A_i$. Then by definition of $A_n$, there exists some $k$ such that $y = i + 5k$. We should prove that $y in A_j$. Add and substract 5 from the right hand side:
$$y = i + 5k + 5 -5$$
$$y = i + 5 + 5(k - 1)$$
Now let $j = i + 1$. Then $y in A_j$ for $j = i + 5$ and $q = k + 1$.
Proof of P2:
Let $n in mathbb{Z}$. We must prove that $n in A_i$ for some $i in I$. But this can be easily shown to be true by the quotient-reminder theorem.
proof-verification elementary-set-theory proof-writing
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
asked Nov 17 at 13:36
amin kamali
11
If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
This is the first exercise of section 3.3 from A Book of Set Theory by Charles C. Pinter.
This is my attempt to solve it, please tell me where I'm doing wrong:
To prove that $A_n$ is a partition of $mathbb{Z}$, we must prove that:
P1: $$If:exists x in A_i cap A_j, then : A_i=A_j$$
P2: $$If :x in A, then: x in A_i :for: some: i in I$$
Suppose $A_i cap A_j$ is a non-empty set. We must prove that $A_i = A_j$
Let $y in A_i$. Then by definition of $A_n$, there exists some $k$ such that $y = i + 5k$. We should prove that $y in A_j$. Add and substract 5 from the right hand side:
$$y = i + 5k + 5 -5$$
$$y = i + 5 + 5(k - 1)$$
Now let $j = i + 1$. Then $y in A_j$ for $j = i + 5$ and $q = k + 1$.
Proof of P2:
Let $n in mathbb{Z}$. We must prove that $n in A_i$ for some $i in I$. But this can be easily shown to be true by the quotient-reminder theorem.
proof-verification elementary-set-theory proof-writing
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
asked Nov 17 at 13:36
amin kamali
11
This is the first exercise of section 3.3 from A Book of Set Theory by Charles C. Pinter.
This is my attempt to solve it, please tell me where I'm doing wrong:
To prove that $A_n$ is a partition of $mathbb{Z}$, we must prove that:
P1: $$If:exists x in A_i cap A_j, then : A_i=A_j$$
P2: $$If :x in A, then: x in A_i :for: some: i in I$$
Suppose $A_i cap A_j$ is a non-empty set. We must prove that $A_i = A_j$
Let $y in A_i$. Then by definition of $A_n$, there exists some $k$ such that $y = i + 5k$. We should prove that $y in A_j$. Add and substract 5 from the right hand side:
$$y = i + 5k + 5 -5$$
$$y = i + 5 + 5(k - 1)$$
Now let $j = i + 1$. Then $y in A_j$ for $j = i + 5$ and $q = k + 1$.
Proof of P2:
Let $n in mathbb{Z}$. We must prove that $n in A_i$ for some $i in I$. But this can be easily shown to be true by the quotient-reminder theorem.
proof-verification elementary-set-theory proof-writing
proof-verification elementary-set-theory proof-writing
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
asked Nov 17 at 13:36
amin kamali
11
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
asked Nov 17 at 13:36
amin kamali
11
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
edited Nov 17 at 14:18
Asaf Karagila♦
300k32421751
300k32421751
asked Nov 17 at 13:36
amin kamali
11
asked Nov 17 at 13:36
amin kamali
11
asked Nov 17 at 13:36
amin kamali
11
11
If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03
add a comment |
If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03
If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03
If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03
add a comment |
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If you already know about partitions induced by the classes of an equivalence relation then show that congruence is an equivalence relation and $A_n$ is the class of integers $,mequiv npmod{!5} $
– Bill Dubuque
Nov 17 at 15:03