Mixing
Which is the correct notation for denoting Dom(f)? X, {$x∈X$|(x, y)$in f$ for some y}, or {$x ∈ X$ :...
$begingroup$
I'm not sure after reading definitions and explanations about notation of Dom(f) in books. Which is correct notation for denoting Dom(f) between X, {$xin X$|(x, y)$in f$ for some y}, and {$x ∈ X$ : $∃y(y ∈Y$ ∧ (x, y) $∈ f$)}?
3.2 Relations
Let R be a relation from A to B. The domain of the relation R, denoted by Dom(R), is the set of all those $a in A$ such that $aRb$ for some $bin B$ such that $aRb$ for some $a in A$. In symbols,
Dom(R)={$a∈A$| $(a, b)∈R$ for some $b∈B$}
and Im(R)={$b∈B$| $(a, b)∈R$ for some $a∈A$}
(. . .)
3.4 Functions
Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$in f$ and (x, z) $in f$, then y=z.
We shall adhere to the custom of writing f: $Xspace rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,space y) in f$.
Source: Set Theory You-Feng Lin, Shwu-Yeng T.Lin
DEFINITION 4.1.4
Let R ⊆ A B. The domain of R is the set
dom(R) = {$x ∈ A$ : ∃y($y ∈B$ ∧ (x, y) $∈ R$)},
and the range of R is the set
ran(R) = {$y∈ B$: ∃x($x∈A∧(x,space y)∈R$)}.
Source: A First Course in Mathematical Logic and Set Theory by Michael L O'leary
elementary-set-theory
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
$endgroup$
add a comment |
$begingroup$
I'm not sure after reading definitions and explanations about notation of Dom(f) in books. Which is correct notation for denoting Dom(f) between X, {$xin X$|(x, y)$in f$ for some y}, and {$x ∈ X$ : $∃y(y ∈Y$ ∧ (x, y) $∈ f$)}?
3.2 Relations
Let R be a relation from A to B. The domain of the relation R, denoted by Dom(R), is the set of all those $a in A$ such that $aRb$ for some $bin B$ such that $aRb$ for some $a in A$. In symbols,
Dom(R)={$a∈A$| $(a, b)∈R$ for some $b∈B$}
and Im(R)={$b∈B$| $(a, b)∈R$ for some $a∈A$}
(. . .)
3.4 Functions
Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$in f$ and (x, z) $in f$, then y=z.
We shall adhere to the custom of writing f: $Xspace rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,space y) in f$.
Source: Set Theory You-Feng Lin, Shwu-Yeng T.Lin
DEFINITION 4.1.4
Let R ⊆ A B. The domain of R is the set
dom(R) = {$x ∈ A$ : ∃y($y ∈B$ ∧ (x, y) $∈ R$)},
and the range of R is the set
ran(R) = {$y∈ B$: ∃x($x∈A∧(x,space y)∈R$)}.
Source: A First Course in Mathematical Logic and Set Theory by Michael L O'leary
elementary-set-theory
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
$endgroup$
$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31
add a comment |
$begingroup$
I'm not sure after reading definitions and explanations about notation of Dom(f) in books. Which is correct notation for denoting Dom(f) between X, {$xin X$|(x, y)$in f$ for some y}, and {$x ∈ X$ : $∃y(y ∈Y$ ∧ (x, y) $∈ f$)}?
3.2 Relations
Let R be a relation from A to B. The domain of the relation R, denoted by Dom(R), is the set of all those $a in A$ such that $aRb$ for some $bin B$ such that $aRb$ for some $a in A$. In symbols,
Dom(R)={$a∈A$| $(a, b)∈R$ for some $b∈B$}
and Im(R)={$b∈B$| $(a, b)∈R$ for some $a∈A$}
(. . .)
3.4 Functions
Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$in f$ and (x, z) $in f$, then y=z.
We shall adhere to the custom of writing f: $Xspace rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,space y) in f$.
Source: Set Theory You-Feng Lin, Shwu-Yeng T.Lin
DEFINITION 4.1.4
Let R ⊆ A B. The domain of R is the set
dom(R) = {$x ∈ A$ : ∃y($y ∈B$ ∧ (x, y) $∈ R$)},
and the range of R is the set
ran(R) = {$y∈ B$: ∃x($x∈A∧(x,space y)∈R$)}.
Source: A First Course in Mathematical Logic and Set Theory by Michael L O'leary
elementary-set-theory
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
$endgroup$
I'm not sure after reading definitions and explanations about notation of Dom(f) in books. Which is correct notation for denoting Dom(f) between X, {$xin X$|(x, y)$in f$ for some y}, and {$x ∈ X$ : $∃y(y ∈Y$ ∧ (x, y) $∈ f$)}?
3.2 Relations
Let R be a relation from A to B. The domain of the relation R, denoted by Dom(R), is the set of all those $a in A$ such that $aRb$ for some $bin B$ such that $aRb$ for some $a in A$. In symbols,
Dom(R)={$a∈A$| $(a, b)∈R$ for some $b∈B$}
and Im(R)={$b∈B$| $(a, b)∈R$ for some $a∈A$}
(. . .)
3.4 Functions
Definition 8. Let X and Y be sets. A function from X to Y is a triple (f, X, Y), where f is a relation from X to Y satisfying
(a) Dom(f) = X.
(b) If (x, y)$in f$ and (x, z) $in f$, then y=z.
We shall adhere to the custom of writing f: $Xspace rightarrow Y$ instead of (f, X, Y) and $y=f(x)$ instead of $(x,space y) in f$.
Source: Set Theory You-Feng Lin, Shwu-Yeng T.Lin
DEFINITION 4.1.4
Let R ⊆ A B. The domain of R is the set
dom(R) = {$x ∈ A$ : ∃y($y ∈B$ ∧ (x, y) $∈ R$)},
and the range of R is the set
ran(R) = {$y∈ B$: ∃x($x∈A∧(x,space y)∈R$)}.
Source: A First Course in Mathematical Logic and Set Theory by Michael L O'leary
elementary-set-theory
elementary-set-theory
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
asked Apr 13 '16 at 13:25
buzzeebuzzee
72321023
72321023
$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31
add a comment |
$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31
$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31
$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31
add a comment |
1 Answer
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votes
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Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $left{ xin Xmidexists y,(x,y)in fright}
$
(2) $left{ xin Xmidexists y,yin Ywedge (x,y)in fright} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)in f$ implies $yin Y$.In such a context, it does not matter then which definitions you use.
answered Apr 13 '16 at 14:24
user155673user155673
1836
$endgroup$
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
add a comment |
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1 Answer
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$begingroup$
Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $left{ xin Xmidexists y,(x,y)in fright}
$
(2) $left{ xin Xmidexists y,yin Ywedge (x,y)in fright} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)in f$ implies $yin Y$.In such a context, it does not matter then which definitions you use.
answered Apr 13 '16 at 14:24
user155673user155673
1836
$endgroup$
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
add a comment |
$begingroup$
Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $left{ xin Xmidexists y,(x,y)in fright}
$
(2) $left{ xin Xmidexists y,yin Ywedge (x,y)in fright} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)in f$ implies $yin Y$.In such a context, it does not matter then which definitions you use.
answered Apr 13 '16 at 14:24
user155673user155673
1836
$endgroup$
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
add a comment |
$begingroup$
Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $left{ xin Xmidexists y,(x,y)in fright}
$
(2) $left{ xin Xmidexists y,yin Ywedge (x,y)in fright} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)in f$ implies $yin Y$.In such a context, it does not matter then which definitions you use.
answered Apr 13 '16 at 14:24
user155673user155673
1836
$endgroup$
Definitions in mathematics come in many forms. This is a matter of presentation and chosen settings. Furthermore, what is considered a good definition in mathematics is a matter of tradition, usefulness, convenience, and so on. This is perhaps more a matter of adequacy. The bottom point is: It does not matter which variant of a definition you work with as long as the variants are logically equivalent.
In your particular case, you have two variants. When expressed in the same formal language (choose your preferred notation here), they are:
(1) $left{ xin Xmidexists y,(x,y)in fright}
$
(2) $left{ xin Xmidexists y,yin Ywedge (x,y)in fright} $
The second definition requires y to be an element of some set Y, the first does not. The two variants are therefore not logically equivalent. In practice however, definitions are used within a certain context. Definition (2) is likely to be used in a context that has already declared the range of f to be included in set Y, in which case $(x,y)in f$ implies $yin Y$.In such a context, it does not matter then which definitions you use.
answered Apr 13 '16 at 14:24
user155673user155673
1836
answered Apr 13 '16 at 14:24
user155673user155673
1836
answered Apr 13 '16 at 14:24
user155673user155673
1836
answered Apr 13 '16 at 14:24
user155673user155673
1836
1836
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
add a comment |
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
Both definitions given in the question are of your form (2): both specify the set from which the second component of the ordered pair must come.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 0:01
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
@BrianM.Scott. You are right. Both definitions given in the provided references are notational variants of each other and correspond to formalization (2). I was however talking of the difference between the definitions mentioned in the title and the opening paragraph.
$endgroup$
– user155673
Apr 14 '16 at 17:52
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
$begingroup$
Ah, okay; I didn’t even notice that.
$endgroup$
– Brian M. Scott
Apr 14 '16 at 17:59
add a comment |
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$begingroup$
The only difference I see between the two is that one uses words (and thus might be thought of as less formal), while the other rigorously follows the rules of second order logic for set-formation.
$endgroup$
– Simon_Peterson
Apr 13 '16 at 13:31