Extension of a theory
$begingroup$
Given is theory $T$ = {$f(x, y) = f(y, x)$} in language only with binary function symbol $f$, (therefore $x, y$ are variables). For each $T_1$ and $T_2$ decide if they are extensions of theory $T$. If yes, show whether they are conservative extensions.
$T_1$ = {$f(x, x) = x$} in the same language as theory $T$
$T_2$ = {$∀u∀v(f(u, v) = c$} in the same language but with constant $c$
I'd really appreciate if someone could help me with this as I am completely lost. I know that one way is to show that $M$($T_1$) $subseteq$ $M$($T$) but how do I show all of the models?
Thank you!
logic predicate-logic
edited Jan 8 at 21:46
Bernard
124k741118
asked Jan 8 at 21:41
AlexAlex
304
$endgroup$
add a comment |
$begingroup$
Given is theory $T$ = {$f(x, y) = f(y, x)$} in language only with binary function symbol $f$, (therefore $x, y$ are variables). For each $T_1$ and $T_2$ decide if they are extensions of theory $T$. If yes, show whether they are conservative extensions.
$T_1$ = {$f(x, x) = x$} in the same language as theory $T$
$T_2$ = {$∀u∀v(f(u, v) = c$} in the same language but with constant $c$
I'd really appreciate if someone could help me with this as I am completely lost. I know that one way is to show that $M$($T_1$) $subseteq$ $M$($T$) but how do I show all of the models?
Thank you!
logic predicate-logic
edited Jan 8 at 21:46
Bernard
124k741118
asked Jan 8 at 21:41
AlexAlex
304
$endgroup$
add a comment |
$begingroup$
Given is theory $T$ = {$f(x, y) = f(y, x)$} in language only with binary function symbol $f$, (therefore $x, y$ are variables). For each $T_1$ and $T_2$ decide if they are extensions of theory $T$. If yes, show whether they are conservative extensions.
$T_1$ = {$f(x, x) = x$} in the same language as theory $T$
$T_2$ = {$∀u∀v(f(u, v) = c$} in the same language but with constant $c$
I'd really appreciate if someone could help me with this as I am completely lost. I know that one way is to show that $M$($T_1$) $subseteq$ $M$($T$) but how do I show all of the models?
Thank you!
logic predicate-logic
edited Jan 8 at 21:46
Bernard
124k741118
asked Jan 8 at 21:41
AlexAlex
304
$endgroup$
Given is theory $T$ = {$f(x, y) = f(y, x)$} in language only with binary function symbol $f$, (therefore $x, y$ are variables). For each $T_1$ and $T_2$ decide if they are extensions of theory $T$. If yes, show whether they are conservative extensions.
$T_1$ = {$f(x, x) = x$} in the same language as theory $T$
$T_2$ = {$∀u∀v(f(u, v) = c$} in the same language but with constant $c$
I'd really appreciate if someone could help me with this as I am completely lost. I know that one way is to show that $M$($T_1$) $subseteq$ $M$($T$) but how do I show all of the models?
Thank you!
logic predicate-logic
logic predicate-logic
edited Jan 8 at 21:46
Bernard
124k741118
asked Jan 8 at 21:41
AlexAlex
304
edited Jan 8 at 21:46
Bernard
124k741118
asked Jan 8 at 21:41
AlexAlex
304
edited Jan 8 at 21:46
Bernard
124k741118
edited Jan 8 at 21:46
Bernard
124k741118
edited Jan 8 at 21:46
Bernard
124k741118
124k741118
asked Jan 8 at 21:41
AlexAlex
304
asked Jan 8 at 21:41
AlexAlex
304
asked Jan 8 at 21:41
AlexAlex
304
304
add a comment |
add a comment |
1 Answer
1
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$begingroup$
Hints:
- Does every model for $T$ satisfy $f(x, x) = x$?
- Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
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add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
oldest
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active
oldest
votes
$begingroup$
Hints:
- Does every model for $T$ satisfy $f(x, x) = x$?
- Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
$endgroup$
add a comment |
$begingroup$
Hints:
- Does every model for $T$ satisfy $f(x, x) = x$?
- Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
$endgroup$
add a comment |
$begingroup$
Hints:
- Does every model for $T$ satisfy $f(x, x) = x$?
- Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
$endgroup$
Hints:
- Does every model for $T$ satisfy $f(x, x) = x$?
- Does every model for $T$ satisfy $f(u, v) = f(u', v')$ for all $u, u', v, v'$?
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
answered Jan 8 at 22:06
Rob ArthanRob Arthan
29.6k42967
29.6k42967
add a comment |
add a comment |
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