Mixing
Translate this sentence to predicate logic
$begingroup$
The question given asks to translate to predicate logic:
Every positive real number has a unique positive real root.
My solution to this problem is to separate it into the appropriate quantifiers.
C(x) = "Every positive real number x"
S(x) = "x has a positive real root"
the final logic form being: ∀x(C(x)→S(x))
It seems to easy for it to be correct. I need help in confirming my answer.
logic predicate-logic logic-translation
edited Jan 28 at 23:10
Bram28
63.9k44793
asked Jan 26 at 22:32
BotBot
61
$endgroup$
add a comment |
$begingroup$
The question given asks to translate to predicate logic:
Every positive real number has a unique positive real root.
My solution to this problem is to separate it into the appropriate quantifiers.
C(x) = "Every positive real number x"
S(x) = "x has a positive real root"
the final logic form being: ∀x(C(x)→S(x))
It seems to easy for it to be correct. I need help in confirming my answer.
logic predicate-logic logic-translation
edited Jan 28 at 23:10
Bram28
63.9k44793
asked Jan 26 at 22:32
BotBot
61
$endgroup$
add a comment |
$begingroup$
The question given asks to translate to predicate logic:
Every positive real number has a unique positive real root.
My solution to this problem is to separate it into the appropriate quantifiers.
C(x) = "Every positive real number x"
S(x) = "x has a positive real root"
the final logic form being: ∀x(C(x)→S(x))
It seems to easy for it to be correct. I need help in confirming my answer.
logic predicate-logic logic-translation
edited Jan 28 at 23:10
Bram28
63.9k44793
asked Jan 26 at 22:32
BotBot
61
$endgroup$
The question given asks to translate to predicate logic:
Every positive real number has a unique positive real root.
My solution to this problem is to separate it into the appropriate quantifiers.
C(x) = "Every positive real number x"
S(x) = "x has a positive real root"
the final logic form being: ∀x(C(x)→S(x))
It seems to easy for it to be correct. I need help in confirming my answer.
logic predicate-logic logic-translation
logic predicate-logic logic-translation
edited Jan 28 at 23:10
Bram28
63.9k44793
asked Jan 26 at 22:32
BotBot
61
edited Jan 28 at 23:10
Bram28
63.9k44793
asked Jan 26 at 22:32
BotBot
61
edited Jan 28 at 23:10
Bram28
63.9k44793
edited Jan 28 at 23:10
Bram28
63.9k44793
edited Jan 28 at 23:10
Bram28
63.9k44793
63.9k44793
asked Jan 26 at 22:32
BotBot
61
asked Jan 26 at 22:32
BotBot
61
asked Jan 26 at 22:32
BotBot
61
61
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
The formula you have translates to:
"For all $x$, if every positive real number $x$, then $x$ has a positive root".
This does not make sense, and does not correspond to the statement you wish to formalize.
What you want to say is
$$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$
i.e.,
"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.
which is equivalent to
"Every positive real number has a unique positive root."
answered Jan 27 at 0:53
MetricMetric
1,22659
$endgroup$
add a comment |
$begingroup$
Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positive real root"
The obvious thing to do is to define:
$C(x)$ = "$x$ is a positive real number"
Second, if we now look at
$$forall x (C(x) rightarrow S(x))$$
then we see that this means: "Every positive real number has a positive real root"
.. but you were supposed to translate "Every positive real number has a unique positive real root"
So, you're missing the "unique" part.
What to do? One thing you can do is to simply redefine:
$S(x)$ = "$x$ has a unique positive real root"
But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$
As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:
$H(x,y)$: "$x$ has $y$ as a positive real root"
OK, so now we can nicely re-express that "Every positive real number has a positive real root":
$$forall x (C(x) rightarrow exists y H(x,y))$$
Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:
$$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$
See how that works? There is a positive real root $y$, but there is not other possible real root $z$
Another way to think about this is that every real root will have to be $y$. So:
$$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$
That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.
Finally, though a little less intuitive, you can do:
$$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$
This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
$endgroup$
add a comment |
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2 Answers
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2 Answers
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$begingroup$
The formula you have translates to:
"For all $x$, if every positive real number $x$, then $x$ has a positive root".
This does not make sense, and does not correspond to the statement you wish to formalize.
What you want to say is
$$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$
i.e.,
"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.
which is equivalent to
"Every positive real number has a unique positive root."
answered Jan 27 at 0:53
MetricMetric
1,22659
$endgroup$
add a comment |
$begingroup$
The formula you have translates to:
"For all $x$, if every positive real number $x$, then $x$ has a positive root".
This does not make sense, and does not correspond to the statement you wish to formalize.
What you want to say is
$$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$
i.e.,
"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.
which is equivalent to
"Every positive real number has a unique positive root."
answered Jan 27 at 0:53
MetricMetric
1,22659
$endgroup$
add a comment |
$begingroup$
The formula you have translates to:
"For all $x$, if every positive real number $x$, then $x$ has a positive root".
This does not make sense, and does not correspond to the statement you wish to formalize.
What you want to say is
$$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$
i.e.,
"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.
which is equivalent to
"Every positive real number has a unique positive root."
answered Jan 27 at 0:53
MetricMetric
1,22659
$endgroup$
The formula you have translates to:
"For all $x$, if every positive real number $x$, then $x$ has a positive root".
This does not make sense, and does not correspond to the statement you wish to formalize.
What you want to say is
$$forall x(text{$x$ is a positive real number} rightarrow text{$x$ has a unique positive root})$$
i.e.,
"For all $x$, if $x$ is a positive real number, then $x$ has a unique positive root.
which is equivalent to
"Every positive real number has a unique positive root."
answered Jan 27 at 0:53
MetricMetric
1,22659
answered Jan 27 at 0:53
MetricMetric
1,22659
answered Jan 27 at 0:53
MetricMetric
1,22659
answered Jan 27 at 0:53
MetricMetric
1,22659
1,22659
add a comment |
add a comment |
$begingroup$
Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positive real root"
The obvious thing to do is to define:
$C(x)$ = "$x$ is a positive real number"
Second, if we now look at
$$forall x (C(x) rightarrow S(x))$$
then we see that this means: "Every positive real number has a positive real root"
.. but you were supposed to translate "Every positive real number has a unique positive real root"
So, you're missing the "unique" part.
What to do? One thing you can do is to simply redefine:
$S(x)$ = "$x$ has a unique positive real root"
But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$
As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:
$H(x,y)$: "$x$ has $y$ as a positive real root"
OK, so now we can nicely re-express that "Every positive real number has a positive real root":
$$forall x (C(x) rightarrow exists y H(x,y))$$
Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:
$$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$
See how that works? There is a positive real root $y$, but there is not other possible real root $z$
Another way to think about this is that every real root will have to be $y$. So:
$$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$
That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.
Finally, though a little less intuitive, you can do:
$$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$
This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
$endgroup$
add a comment |
$begingroup$
Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positive real root"
The obvious thing to do is to define:
$C(x)$ = "$x$ is a positive real number"
Second, if we now look at
$$forall x (C(x) rightarrow S(x))$$
then we see that this means: "Every positive real number has a positive real root"
.. but you were supposed to translate "Every positive real number has a unique positive real root"
So, you're missing the "unique" part.
What to do? One thing you can do is to simply redefine:
$S(x)$ = "$x$ has a unique positive real root"
But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$
As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:
$H(x,y)$: "$x$ has $y$ as a positive real root"
OK, so now we can nicely re-express that "Every positive real number has a positive real root":
$$forall x (C(x) rightarrow exists y H(x,y))$$
Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:
$$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$
See how that works? There is a positive real root $y$, but there is not other possible real root $z$
Another way to think about this is that every real root will have to be $y$. So:
$$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$
That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.
Finally, though a little less intuitive, you can do:
$$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$
This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
$endgroup$
add a comment |
$begingroup$
Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positive real root"
The obvious thing to do is to define:
$C(x)$ = "$x$ is a positive real number"
Second, if we now look at
$$forall x (C(x) rightarrow S(x))$$
then we see that this means: "Every positive real number has a positive real root"
.. but you were supposed to translate "Every positive real number has a unique positive real root"
So, you're missing the "unique" part.
What to do? One thing you can do is to simply redefine:
$S(x)$ = "$x$ has a unique positive real root"
But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$
As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:
$H(x,y)$: "$x$ has $y$ as a positive real root"
OK, so now we can nicely re-express that "Every positive real number has a positive real root":
$$forall x (C(x) rightarrow exists y H(x,y))$$
Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:
$$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$
See how that works? There is a positive real root $y$, but there is not other possible real root $z$
Another way to think about this is that every real root will have to be $y$. So:
$$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$
That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.
Finally, though a little less intuitive, you can do:
$$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$
This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
$endgroup$
Couple of things:
First, you don't want to define:
$C(x)$ = "Every positive real number $x$"
The 'every' needs to be taken care of by a quantifier so that should not be part of the meaning of this formula. Indeed, the formula $C(x)$ should simply express a property of $x$, just like your:
$S(x)$ = "$x$ has a positive real root"
The obvious thing to do is to define:
$C(x)$ = "$x$ is a positive real number"
Second, if we now look at
$$forall x (C(x) rightarrow S(x))$$
then we see that this means: "Every positive real number has a positive real root"
.. but you were supposed to translate "Every positive real number has a unique positive real root"
So, you're missing the "unique" part.
What to do? One thing you can do is to simply redefine:
$S(x)$ = "$x$ has a unique positive real root"
But given as you are using predicate logic, I think it is much better to further analyze this into objects and predicates. In particular, that "unique positive real root" is an object itself, so it makes sense to use a (separate) variable for it, say $y$
As such, instead of having a 1-place predicate $S(x)$ that says that "x has a unique positive real root", you want to use a 2-place predicate:
$H(x,y)$: "$x$ has $y$ as a positive real root"
OK, so now we can nicely re-express that "Every positive real number has a positive real root":
$$forall x (C(x) rightarrow exists y H(x,y))$$
Hmm, but we still leave out the "uniqueness" part. Well, the cool thing is that in predicate logic you can capture that using the identity relationship $=$. That is, you want to say that not only does there exist some $y$ that is a positive real root of $x$, but that this $y$ is the only positive real root of $x$, i.e. that there are no other positive real roots of $x$. As such, we can write:
$$forall x (C(x) rightarrow exists y (H(x,y) land neg exists z (H(x,z) land z not = y)))$$
See how that works? There is a positive real root $y$, but there is not other possible real root $z$
Another way to think about this is that every real root will have to be $y$. So:
$$forall x (C(x) rightarrow exists y (H(x,y) land forall z (H(x,z) rightarrow z = y)))$$
That is: there is a positive real root $y$, and everything that is a positive real root will have to be $y$, thus making $y$ the one and only one positive real root.
Finally, though a little less intuitive, you can do:
$$forall x (C(x) rightarrow exists y forall z (H(x,z) leftrightarrow z = y))$$
This is equivalent to the last one, because $y$ is of course equal to $y$, so if tyou set $z$ to $y$, then you can go from right to left, and hence $z=y$ becomes a positive real root of $x$, but going left to right, we again get that any positive real root will have to be $y$
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
answered Jan 28 at 16:18
Bram28Bram28
63.9k44793
63.9k44793
add a comment |
add a comment |
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