Mixing
Definition of a structure
$begingroup$
I found the following definition for a structure in my math course:
A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $r$:
-The value $c^x$ for an object-symbol $c$ is an element of the universum $D_x$. $c$ can be a constant value or a variable.
-The value $F^x$ for a function-symbol $F/n$ is a function $F^x:D_x^n rightarrow D_x$. This is a function that maps n-values $(a_1,...,a_n)$ from the universum to single values of the universum.
-The value $P^x$ for a predicate-symbol $P/n$ is a n-relation $P^x$ in $D_x$, so $P^x subset D_x^n$.
We call $r^x$ the value or interpretation of $r$ in X.
=================================================
This seems pretty abstract to me,
Can someone give me an example of a structure with this definition?
Or just more information/explanation?
I looked on the internet but didn't find anything.
logic definition examples-counterexamples predicate-logic
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
add a comment |
$begingroup$
I found the following definition for a structure in my math course:
A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $r$:
-The value $c^x$ for an object-symbol $c$ is an element of the universum $D_x$. $c$ can be a constant value or a variable.
-The value $F^x$ for a function-symbol $F/n$ is a function $F^x:D_x^n rightarrow D_x$. This is a function that maps n-values $(a_1,...,a_n)$ from the universum to single values of the universum.
-The value $P^x$ for a predicate-symbol $P/n$ is a n-relation $P^x$ in $D_x$, so $P^x subset D_x^n$.
We call $r^x$ the value or interpretation of $r$ in X.
=================================================
This seems pretty abstract to me,
Can someone give me an example of a structure with this definition?
Or just more information/explanation?
I looked on the internet but didn't find anything.
logic definition examples-counterexamples predicate-logic
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
add a comment |
$begingroup$
I found the following definition for a structure in my math course:
A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $r$:
-The value $c^x$ for an object-symbol $c$ is an element of the universum $D_x$. $c$ can be a constant value or a variable.
-The value $F^x$ for a function-symbol $F/n$ is a function $F^x:D_x^n rightarrow D_x$. This is a function that maps n-values $(a_1,...,a_n)$ from the universum to single values of the universum.
-The value $P^x$ for a predicate-symbol $P/n$ is a n-relation $P^x$ in $D_x$, so $P^x subset D_x^n$.
We call $r^x$ the value or interpretation of $r$ in X.
=================================================
This seems pretty abstract to me,
Can someone give me an example of a structure with this definition?
Or just more information/explanation?
I looked on the internet but didn't find anything.
logic definition examples-counterexamples predicate-logic
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
$endgroup$
I found the following definition for a structure in my math course:
A structure $X$ consists of a non-empty set $D_x$, the universum of $X$ and the attribution of values $r^x$ to non logical-symbols $r$:
-The value $c^x$ for an object-symbol $c$ is an element of the universum $D_x$. $c$ can be a constant value or a variable.
-The value $F^x$ for a function-symbol $F/n$ is a function $F^x:D_x^n rightarrow D_x$. This is a function that maps n-values $(a_1,...,a_n)$ from the universum to single values of the universum.
-The value $P^x$ for a predicate-symbol $P/n$ is a n-relation $P^x$ in $D_x$, so $P^x subset D_x^n$.
We call $r^x$ the value or interpretation of $r$ in X.
=================================================
This seems pretty abstract to me,
Can someone give me an example of a structure with this definition?
Or just more information/explanation?
I looked on the internet but didn't find anything.
logic definition examples-counterexamples predicate-logic
logic definition examples-counterexamples predicate-logic
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
edited Jan 23 at 14:45
GNUSupporter 8964民主女神 地下教會
14k82650
14k82650
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
asked Jan 23 at 14:40
Ayoub RossiAyoub Rossi
11110
11110
add a comment |
add a comment |
1 Answer
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$begingroup$
You can see Peano arithmetic and the corresponding structure of natural numbers :
$(mathbb N, 0, S, +, times)$.
Here $mathbb N = { 0,1,2,ldots }$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.
$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $text {Succ} : mathbb N to mathbb N$.
Finally, $+$ and $times$ are binary function symbols, interpreted with sum and product respectively.
If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $text {less} subseteq mathbb N times mathbb N$.
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
$endgroup$
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
add a comment |
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1 Answer
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1 Answer
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$begingroup$
You can see Peano arithmetic and the corresponding structure of natural numbers :
$(mathbb N, 0, S, +, times)$.
Here $mathbb N = { 0,1,2,ldots }$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.
$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $text {Succ} : mathbb N to mathbb N$.
Finally, $+$ and $times$ are binary function symbols, interpreted with sum and product respectively.
If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $text {less} subseteq mathbb N times mathbb N$.
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
$endgroup$
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
add a comment |
$begingroup$
You can see Peano arithmetic and the corresponding structure of natural numbers :
$(mathbb N, 0, S, +, times)$.
Here $mathbb N = { 0,1,2,ldots }$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.
$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $text {Succ} : mathbb N to mathbb N$.
Finally, $+$ and $times$ are binary function symbols, interpreted with sum and product respectively.
If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $text {less} subseteq mathbb N times mathbb N$.
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
$endgroup$
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
add a comment |
$begingroup$
You can see Peano arithmetic and the corresponding structure of natural numbers :
$(mathbb N, 0, S, +, times)$.
Here $mathbb N = { 0,1,2,ldots }$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.
$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $text {Succ} : mathbb N to mathbb N$.
Finally, $+$ and $times$ are binary function symbols, interpreted with sum and product respectively.
If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $text {less} subseteq mathbb N times mathbb N$.
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
$endgroup$
You can see Peano arithmetic and the corresponding structure of natural numbers :
$(mathbb N, 0, S, +, times)$.
Here $mathbb N = { 0,1,2,ldots }$ is the domain $D$ and $0$ is the only individual constant $c$ denoting the number zero.
$S(x)$ is a function symbol that is interpreted with the successor function, i.e. $text {Succ} : mathbb N to mathbb N$.
Finally, $+$ and $times$ are binary function symbols, interpreted with sum and product respectively.
If we want to consider also an example of relation, we have to consider the binary predicate symbol $<$, that will be interpreted with the "less then" relation, i.e. $text {less} subseteq mathbb N times mathbb N$.
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
answered Jan 23 at 14:53
Mauro ALLEGRANZAMauro ALLEGRANZA
67.1k449115
67.1k449115
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
add a comment |
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
Is this an example of a first-order structure? If yes, does the "n" in my definition stands for n-th order or not really?
$endgroup$
– Ayoub Rossi
Jan 23 at 16:19
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
$begingroup$
@AyoubRossi - Yes, it is a FO structure because I've assume FO Logic. What is $n$ ? Is it in $F/n$ ? if so, it means the arity (i.e. the number of argument places) of the symbol. Usually is $F_n$ meaning $F(x_1,x_2, ldots, x_n)$. If so, $S(x)$ has arity $1$; while $+,times$ and $<$ all have arity $2$ (they are binary functions/relations).
$endgroup$
– Mauro ALLEGRANZA
Jan 23 at 16:25
add a comment |
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