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Set theory - Axiom of Union [duplicate]
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Doubts on the axiom of union?
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Reading Comprehensive Mathematics for Computer Scientists 1 chapter 2:
Axiom 3 (Axiom of Union) If a is a set, then there is a set:
there exists an element b∈a such that x∈b.
This set is denoted by ⋃a and is called the union of a.
Notation 2 If a = b,c. or a = b,c,d, respectively, one also writes b ∪ c, or b ∪ c ∪ d, respectively, instead of ∪a
What if a = 1,2 then b = 1 how can x belong to b if it's not a set?
Sorry I am a little confused...
Thanks in advance
elementary-set-theory
asked Jul 24 at 5:16
Paul
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marked as duplicate by Asaf Karagila elementary-set-theory
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Jul 24 at 11:41
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This question already has an answer here:
Doubts on the axiom of union?
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Reading Comprehensive Mathematics for Computer Scientists 1 chapter 2:
Axiom 3 (Axiom of Union) If a is a set, then there is a set:
there exists an element b∈a such that x∈b.
This set is denoted by ⋃a and is called the union of a.
Notation 2 If a = b,c. or a = b,c,d, respectively, one also writes b ∪ c, or b ∪ c ∪ d, respectively, instead of ∪a
What if a = 1,2 then b = 1 how can x belong to b if it's not a set?
Sorry I am a little confused...
Thanks in advance
elementary-set-theory
asked Jul 24 at 5:16
Paul
135
marked as duplicate by Asaf Karagila elementary-set-theory
Users with the  elementary-set-theory badge can single-handedly close elementary-set-theory questions as duplicates and reopen them as needed.
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Jul 24 at 11:41
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
2
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
That's correct.
– Bungo
Jul 24 at 5:24
Thanks for your help
– Paul
Jul 24 at 5:28
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40
add a comment |Â
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up vote
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This question already has an answer here:
Doubts on the axiom of union?
1 answer
Reading Comprehensive Mathematics for Computer Scientists 1 chapter 2:
Axiom 3 (Axiom of Union) If a is a set, then there is a set:
there exists an element b∈a such that x∈b.
This set is denoted by ⋃a and is called the union of a.
Notation 2 If a = b,c. or a = b,c,d, respectively, one also writes b ∪ c, or b ∪ c ∪ d, respectively, instead of ∪a
What if a = 1,2 then b = 1 how can x belong to b if it's not a set?
Sorry I am a little confused...
Thanks in advance
elementary-set-theory
asked Jul 24 at 5:16
Paul
135
This question already has an answer here:
Doubts on the axiom of union?
1 answer
Reading Comprehensive Mathematics for Computer Scientists 1 chapter 2:
Axiom 3 (Axiom of Union) If a is a set, then there is a set:
there exists an element b∈a such that x∈b.
This set is denoted by ⋃a and is called the union of a.
Notation 2 If a = b,c. or a = b,c,d, respectively, one also writes b ∪ c, or b ∪ c ∪ d, respectively, instead of ∪a
What if a = 1,2 then b = 1 how can x belong to b if it's not a set?
Sorry I am a little confused...
Thanks in advance
This question already has an answer here:
Doubts on the axiom of union?
1 answer
elementary-set-theory
asked Jul 24 at 5:16
Paul
135
asked Jul 24 at 5:16
Paul
135
asked Jul 24 at 5:16
Paul
135
asked Jul 24 at 5:16
Paul
135
135
marked as duplicate by Asaf Karagila elementary-set-theory
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Jul 24 at 11:41
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
marked as duplicate by Asaf Karagila elementary-set-theory
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Jul 24 at 11:41
This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.
2
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
That's correct.
– Bungo
Jul 24 at 5:24
Thanks for your help
– Paul
Jul 24 at 5:28
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40
add a comment |Â
2
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
That's correct.
– Bungo
Jul 24 at 5:24
Thanks for your help
– Paul
Jul 24 at 5:28
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40
2
2
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
That's correct.
– Bungo
Jul 24 at 5:24
That's correct.
– Bungo
Jul 24 at 5:24
Thanks for your help
– Paul
Jul 24 at 5:28
Thanks for your help
– Paul
Jul 24 at 5:28
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40
add a comment |Â
1 Answer
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Usually, in frameworks where the axiom of union appears (e.g. ZFC), everything is a set !
For instance, the natural numbers are usually defined to be $0:=$ and recursively by $n+1 := ncup n$. More informally, we have $n= 0,..., n-1$.
So for instance, in your example
$$bigcup 1,2 = bigcup 0, 0,1 = 0,1$$
more generally it is easy to see that if $A$ is a finite set consisting only of natural numbers (with the above definition), then $bigcup A = max A$.
answered Jul 24 at 11:37
Tashi Walde
1,20910
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1 Answer
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1 Answer
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active
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active
oldest
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active
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up vote
1
down vote
Usually, in frameworks where the axiom of union appears (e.g. ZFC), everything is a set !
For instance, the natural numbers are usually defined to be $0:=$ and recursively by $n+1 := ncup n$. More informally, we have $n= 0,..., n-1$.
So for instance, in your example
$$bigcup 1,2 = bigcup 0, 0,1 = 0,1$$
more generally it is easy to see that if $A$ is a finite set consisting only of natural numbers (with the above definition), then $bigcup A = max A$.
answered Jul 24 at 11:37
Tashi Walde
1,20910
add a comment |Â
up vote
1
down vote
Usually, in frameworks where the axiom of union appears (e.g. ZFC), everything is a set !
For instance, the natural numbers are usually defined to be $0:=$ and recursively by $n+1 := ncup n$. More informally, we have $n= 0,..., n-1$.
So for instance, in your example
$$bigcup 1,2 = bigcup 0, 0,1 = 0,1$$
more generally it is easy to see that if $A$ is a finite set consisting only of natural numbers (with the above definition), then $bigcup A = max A$.
answered Jul 24 at 11:37
Tashi Walde
1,20910
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Usually, in frameworks where the axiom of union appears (e.g. ZFC), everything is a set !
For instance, the natural numbers are usually defined to be $0:=$ and recursively by $n+1 := ncup n$. More informally, we have $n= 0,..., n-1$.
So for instance, in your example
$$bigcup 1,2 = bigcup 0, 0,1 = 0,1$$
more generally it is easy to see that if $A$ is a finite set consisting only of natural numbers (with the above definition), then $bigcup A = max A$.
answered Jul 24 at 11:37
Tashi Walde
1,20910
Usually, in frameworks where the axiom of union appears (e.g. ZFC), everything is a set !
For instance, the natural numbers are usually defined to be $0:=$ and recursively by $n+1 := ncup n$. More informally, we have $n= 0,..., n-1$.
So for instance, in your example
$$bigcup 1,2 = bigcup 0, 0,1 = 0,1$$
more generally it is easy to see that if $A$ is a finite set consisting only of natural numbers (with the above definition), then $bigcup A = max A$.
answered Jul 24 at 11:37
Tashi Walde
1,20910
answered Jul 24 at 11:37
Tashi Walde
1,20910
answered Jul 24 at 11:37
Tashi Walde
1,20910
answered Jul 24 at 11:37
Tashi Walde
1,20910
1,20910
add a comment |Â
add a comment |Â
2
In your example, $cup a$ would be the empty set, because there is no $x$ satisfying the definition. For a more interesting example, consider $a = 1, 2$.
– Bungo
Jul 24 at 5:21
Okay thanks, so in your example Ua = 1,2?
– Paul
Jul 24 at 5:23
That's correct.
– Bungo
Jul 24 at 5:24
Thanks for your help
– Paul
Jul 24 at 5:28
@Bungo, you might want to be a bit careful. For instance, the natural number $n$ is often defined to be the set $0,...,n-1$, in which case what you are saying is not true. See my answer
– Tashi Walde
Jul 24 at 11:40