Given $f : A rightarrow B :::forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$.
Suppose that $A$ and $B$ are nonempty subsets of $mathbb{R}$, and that $f : A rightarrow B$ is a function satisfying $forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$.
(Such a function is said to be weakly increasing.) Furthermore, suppose that $f$ is surjective
(i) Prove that if $M$ is a largest element of $A$, then $f(M)$ is a largest element of $B$.
(ii) Why is the assumption that $f$ is surjective necessary? Give an example of nonempty subsets $A$ and $B$
of $mathbb{R}$, a weakly increasing function $f : A rightarrow B$, and a largest element $M$ of $A$ such that $f(M)$ is not a
largest element of $B$.
real-analysis functions
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
add a comment |
Suppose that $A$ and $B$ are nonempty subsets of $mathbb{R}$, and that $f : A rightarrow B$ is a function satisfying $forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$.
(Such a function is said to be weakly increasing.) Furthermore, suppose that $f$ is surjective
(i) Prove that if $M$ is a largest element of $A$, then $f(M)$ is a largest element of $B$.
(ii) Why is the assumption that $f$ is surjective necessary? Give an example of nonempty subsets $A$ and $B$
of $mathbb{R}$, a weakly increasing function $f : A rightarrow B$, and a largest element $M$ of $A$ such that $f(M)$ is not a
largest element of $B$.
real-analysis functions
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55
add a comment |
Suppose that $A$ and $B$ are nonempty subsets of $mathbb{R}$, and that $f : A rightarrow B$ is a function satisfying $forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$.
(Such a function is said to be weakly increasing.) Furthermore, suppose that $f$ is surjective
(i) Prove that if $M$ is a largest element of $A$, then $f(M)$ is a largest element of $B$.
(ii) Why is the assumption that $f$ is surjective necessary? Give an example of nonempty subsets $A$ and $B$
of $mathbb{R}$, a weakly increasing function $f : A rightarrow B$, and a largest element $M$ of $A$ such that $f(M)$ is not a
largest element of $B$.
real-analysis functions
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
Suppose that $A$ and $B$ are nonempty subsets of $mathbb{R}$, and that $f : A rightarrow B$ is a function satisfying $forall: a_1, a_2 in A,:a_1 leq a_2 implies f(a_1) leq f(a_2)$.
(Such a function is said to be weakly increasing.) Furthermore, suppose that $f$ is surjective
(i) Prove that if $M$ is a largest element of $A$, then $f(M)$ is a largest element of $B$.
(ii) Why is the assumption that $f$ is surjective necessary? Give an example of nonempty subsets $A$ and $B$
of $mathbb{R}$, a weakly increasing function $f : A rightarrow B$, and a largest element $M$ of $A$ such that $f(M)$ is not a
largest element of $B$.
real-analysis functions
real-analysis functions
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
edited Nov 22 '18 at 17:57
Yadati Kiran
1,692619
1,692619
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
asked Nov 22 '18 at 17:40
jennifer okonkwojennifer okonkwo
93
93
I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55
add a comment |
I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55
I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55
add a comment |
1 Answer
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Counter-example in (ii):
Let $A={0}$ and $B={0,1}$. Define $f:Arightarrow B$ by $f(0)=0$. Clearly, $A,B$ are non-empty subsets of $mathbb{R}$ and $f$ is "weaking increasing". Moreover $0$ is the largest element of $A$. However, $f(0)=0$ is not the largest element of $B$.
//////////////////////////////////
To prove (i). Let $yin B$. Since $f$ is surjective, there exists $xin A$ such that $f(x)=y$. Now $xleq M$, so $y=f(x)leq f(M)$. This shows that $f(M)$ is the largest element in $B$.
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
add a comment |
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Counter-example in (ii):
Let $A={0}$ and $B={0,1}$. Define $f:Arightarrow B$ by $f(0)=0$. Clearly, $A,B$ are non-empty subsets of $mathbb{R}$ and $f$ is "weaking increasing". Moreover $0$ is the largest element of $A$. However, $f(0)=0$ is not the largest element of $B$.
//////////////////////////////////
To prove (i). Let $yin B$. Since $f$ is surjective, there exists $xin A$ such that $f(x)=y$. Now $xleq M$, so $y=f(x)leq f(M)$. This shows that $f(M)$ is the largest element in $B$.
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
add a comment |
Counter-example in (ii):
Let $A={0}$ and $B={0,1}$. Define $f:Arightarrow B$ by $f(0)=0$. Clearly, $A,B$ are non-empty subsets of $mathbb{R}$ and $f$ is "weaking increasing". Moreover $0$ is the largest element of $A$. However, $f(0)=0$ is not the largest element of $B$.
//////////////////////////////////
To prove (i). Let $yin B$. Since $f$ is surjective, there exists $xin A$ such that $f(x)=y$. Now $xleq M$, so $y=f(x)leq f(M)$. This shows that $f(M)$ is the largest element in $B$.
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
add a comment |
Counter-example in (ii):
Let $A={0}$ and $B={0,1}$. Define $f:Arightarrow B$ by $f(0)=0$. Clearly, $A,B$ are non-empty subsets of $mathbb{R}$ and $f$ is "weaking increasing". Moreover $0$ is the largest element of $A$. However, $f(0)=0$ is not the largest element of $B$.
//////////////////////////////////
To prove (i). Let $yin B$. Since $f$ is surjective, there exists $xin A$ such that $f(x)=y$. Now $xleq M$, so $y=f(x)leq f(M)$. This shows that $f(M)$ is the largest element in $B$.
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
Counter-example in (ii):
Let $A={0}$ and $B={0,1}$. Define $f:Arightarrow B$ by $f(0)=0$. Clearly, $A,B$ are non-empty subsets of $mathbb{R}$ and $f$ is "weaking increasing". Moreover $0$ is the largest element of $A$. However, $f(0)=0$ is not the largest element of $B$.
//////////////////////////////////
To prove (i). Let $yin B$. Since $f$ is surjective, there exists $xin A$ such that $f(x)=y$. Now $xleq M$, so $y=f(x)leq f(M)$. This shows that $f(M)$ is the largest element in $B$.
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
answered Nov 22 '18 at 17:51
Danny Pak-Keung ChanDanny Pak-Keung Chan
2,22538
2,22538
add a comment |
add a comment |
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I've taken the liberty of making some formatting adjustments to your post, fixing the tags, and trying to make the title more helpful. I'm not certain what "a largest element" means. Do you mean "the greatest element," possibly? Also, your questions will be more likely to gain attention if you include your thoughts and efforts, to help us gauge your experience level.
– Cameron Buie
Nov 22 '18 at 17:53
For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to.
– Cameron Buie
Nov 22 '18 at 17:55
Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. Welcome to Math.SE!
– Cameron Buie
Nov 22 '18 at 17:55