Determine whether $S cap T$ is closed and bounded
$begingroup$
Let $$S={xin mathbb{R} : text{ } x^6-x^5 leq 100}$$
$$T={x^2-2x : text{ } xin (0, infty)}$$
Determine wether $ S cap T$ is closed and bounded?
My Try
$S$ is bounded hence $ S cap T$ is also bounded.
$S=[a,b]$ where $a<-1$ because -1 satisfy the equation given. and $T=(-1, infty)$ hence $ S cap T=[-1,b]$
Hence $ S cap T$ is closed and bounded
However I am not satisfied with my reasoning.
Can anyone tell me the proper solution.
real-analysis calculus analysis elementary-set-theory
$endgroup$
add a comment |
$begingroup$
Let $$S={xin mathbb{R} : text{ } x^6-x^5 leq 100}$$
$$T={x^2-2x : text{ } xin (0, infty)}$$
Determine wether $ S cap T$ is closed and bounded?
My Try
$S$ is bounded hence $ S cap T$ is also bounded.
$S=[a,b]$ where $a<-1$ because -1 satisfy the equation given. and $T=(-1, infty)$ hence $ S cap T=[-1,b]$
Hence $ S cap T$ is closed and bounded
However I am not satisfied with my reasoning.
Can anyone tell me the proper solution.
real-analysis calculus analysis elementary-set-theory
$endgroup$
$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
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In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
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– Christoph
Dec 18 '18 at 15:49
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
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@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21
add a comment |
$begingroup$
Let $$S={xin mathbb{R} : text{ } x^6-x^5 leq 100}$$
$$T={x^2-2x : text{ } xin (0, infty)}$$
Determine wether $ S cap T$ is closed and bounded?
My Try
$S$ is bounded hence $ S cap T$ is also bounded.
$S=[a,b]$ where $a<-1$ because -1 satisfy the equation given. and $T=(-1, infty)$ hence $ S cap T=[-1,b]$
Hence $ S cap T$ is closed and bounded
However I am not satisfied with my reasoning.
Can anyone tell me the proper solution.
real-analysis calculus analysis elementary-set-theory
$endgroup$
Let $$S={xin mathbb{R} : text{ } x^6-x^5 leq 100}$$
$$T={x^2-2x : text{ } xin (0, infty)}$$
Determine wether $ S cap T$ is closed and bounded?
My Try
$S$ is bounded hence $ S cap T$ is also bounded.
$S=[a,b]$ where $a<-1$ because -1 satisfy the equation given. and $T=(-1, infty)$ hence $ S cap T=[-1,b]$
Hence $ S cap T$ is closed and bounded
However I am not satisfied with my reasoning.
Can anyone tell me the proper solution.
real-analysis calculus analysis elementary-set-theory
real-analysis calculus analysis elementary-set-theory
edited Dec 18 '18 at 16:17
Rakesh Bhatt
asked Dec 18 '18 at 15:46
Rakesh BhattRakesh Bhatt
952214
952214
$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
$begingroup$
In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
$endgroup$
– Christoph
Dec 18 '18 at 15:49
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
$begingroup$
@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21
add a comment |
$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
$begingroup$
In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
$endgroup$
– Christoph
Dec 18 '18 at 15:49
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
$begingroup$
@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21
$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
$begingroup$
In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
$endgroup$
– Christoph
Dec 18 '18 at 15:49
$begingroup$
In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
$endgroup$
– Christoph
Dec 18 '18 at 15:49
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
$begingroup$
@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Let $s(x) = x^6-x^5$, since $s$ is continuous, $S=s^{-1} ((-infty, 100])$ is closed.
Since $s(x) to infty$ as $|x| to infty$ we see that $S$ is bounded, hence compact.
Let $t(x) = x^2-2x = x(x-2)$, differentiating shows that $t$ has a $min$ of $-1$. at $x=1$,
and we see that $t(x) to infty$ as $x to infty$. Hence $T=[-1,infty)$, which is
closed.
Hence $T cap S$ is closed.
$endgroup$
add a comment |
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1 Answer
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1 Answer
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oldest
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$begingroup$
Let $s(x) = x^6-x^5$, since $s$ is continuous, $S=s^{-1} ((-infty, 100])$ is closed.
Since $s(x) to infty$ as $|x| to infty$ we see that $S$ is bounded, hence compact.
Let $t(x) = x^2-2x = x(x-2)$, differentiating shows that $t$ has a $min$ of $-1$. at $x=1$,
and we see that $t(x) to infty$ as $x to infty$. Hence $T=[-1,infty)$, which is
closed.
Hence $T cap S$ is closed.
$endgroup$
add a comment |
$begingroup$
Let $s(x) = x^6-x^5$, since $s$ is continuous, $S=s^{-1} ((-infty, 100])$ is closed.
Since $s(x) to infty$ as $|x| to infty$ we see that $S$ is bounded, hence compact.
Let $t(x) = x^2-2x = x(x-2)$, differentiating shows that $t$ has a $min$ of $-1$. at $x=1$,
and we see that $t(x) to infty$ as $x to infty$. Hence $T=[-1,infty)$, which is
closed.
Hence $T cap S$ is closed.
$endgroup$
add a comment |
$begingroup$
Let $s(x) = x^6-x^5$, since $s$ is continuous, $S=s^{-1} ((-infty, 100])$ is closed.
Since $s(x) to infty$ as $|x| to infty$ we see that $S$ is bounded, hence compact.
Let $t(x) = x^2-2x = x(x-2)$, differentiating shows that $t$ has a $min$ of $-1$. at $x=1$,
and we see that $t(x) to infty$ as $x to infty$. Hence $T=[-1,infty)$, which is
closed.
Hence $T cap S$ is closed.
$endgroup$
Let $s(x) = x^6-x^5$, since $s$ is continuous, $S=s^{-1} ((-infty, 100])$ is closed.
Since $s(x) to infty$ as $|x| to infty$ we see that $S$ is bounded, hence compact.
Let $t(x) = x^2-2x = x(x-2)$, differentiating shows that $t$ has a $min$ of $-1$. at $x=1$,
and we see that $t(x) to infty$ as $x to infty$. Hence $T=[-1,infty)$, which is
closed.
Hence $T cap S$ is closed.
answered Dec 18 '18 at 16:26
copper.hatcopper.hat
127k559160
127k559160
add a comment |
add a comment |
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$begingroup$
Why is $T=(-2,infty)$? And if it were the case, would there not be $S cap T=(-2,b]$?
$endgroup$
– Mindlack
Dec 18 '18 at 15:49
$begingroup$
In the title you ask for open or closed, in the body you ask for closed and bounded. Which is it?
$endgroup$
– Christoph
Dec 18 '18 at 15:49
$begingroup$
Note that $T=[-1,infty)$.
$endgroup$
– copper.hat
Dec 18 '18 at 16:10
$begingroup$
@cristoph Its closed and bounded
$endgroup$
– Rakesh Bhatt
Dec 18 '18 at 16:19
$begingroup$
You have to justify why $S$ is a bounded interval, and your formula for $T$ is incorrect, take $x=1$ in the defining set.
$endgroup$
– copper.hat
Dec 18 '18 at 16:21