Understanding upper and lower bounds (and supr and infin)
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I am following a course in discrete mathemathics and I am having great dificulty in understanding the lower, upper bounds and its lub's and glb's. This keeps me from fully understanding everything else.
Consider the set $P = {xmid xinmathbb{R} , 1 leq x < 2}$. What are the upper bounds, lower bounds, greatest lower bounds and least upper bounds?
Here is what i know.
Definition, consider the poset $A$ and its subset $B$. An element $ain A$ is called an upper bound if $bleq a$ for all $bin B$. An element $ain A$ is called an lower bound if $aleq b$ for all $bin B$.
So my first go was, that $1$ is the greatest lower bound with lower bounds $1leq x < 2$ in $P$, and there exists no upperbound since $2$ is not smaller than $2$. And since there is no upper bound, there cannot be a least upper bound.
However, this when I ONLY consider the numbers in $P$ ($P$ is the universe). If i would consider $P$ to be a subset, that is $Psubseteq (mathbb{R}, leq)$, then I would guess the least upper bound is $2$, and all upper bounds are $xin[2,infty)$. Same for lower bounds, $1$ is the greatest lower bound and $xin(-infty, 1]$ are all the lower bounds.
I think the former is totally wrong, since subset definition is not used, and the latter is partially wrong since $2$ is not contained in $P$, but it is included in my upper bounds. However, I am not sure whether or not to consider $P$ to be a subset of anything else in order to give an answer. Are there two ways to look at this problem?
In desperate need for some enlightening.
discrete-mathematics
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
$endgroup$
add a comment |
$begingroup$
I am following a course in discrete mathemathics and I am having great dificulty in understanding the lower, upper bounds and its lub's and glb's. This keeps me from fully understanding everything else.
Consider the set $P = {xmid xinmathbb{R} , 1 leq x < 2}$. What are the upper bounds, lower bounds, greatest lower bounds and least upper bounds?
Here is what i know.
Definition, consider the poset $A$ and its subset $B$. An element $ain A$ is called an upper bound if $bleq a$ for all $bin B$. An element $ain A$ is called an lower bound if $aleq b$ for all $bin B$.
So my first go was, that $1$ is the greatest lower bound with lower bounds $1leq x < 2$ in $P$, and there exists no upperbound since $2$ is not smaller than $2$. And since there is no upper bound, there cannot be a least upper bound.
However, this when I ONLY consider the numbers in $P$ ($P$ is the universe). If i would consider $P$ to be a subset, that is $Psubseteq (mathbb{R}, leq)$, then I would guess the least upper bound is $2$, and all upper bounds are $xin[2,infty)$. Same for lower bounds, $1$ is the greatest lower bound and $xin(-infty, 1]$ are all the lower bounds.
I think the former is totally wrong, since subset definition is not used, and the latter is partially wrong since $2$ is not contained in $P$, but it is included in my upper bounds. However, I am not sure whether or not to consider $P$ to be a subset of anything else in order to give an answer. Are there two ways to look at this problem?
In desperate need for some enlightening.
discrete-mathematics
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
$endgroup$
$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32
add a comment |
$begingroup$
I am following a course in discrete mathemathics and I am having great dificulty in understanding the lower, upper bounds and its lub's and glb's. This keeps me from fully understanding everything else.
Consider the set $P = {xmid xinmathbb{R} , 1 leq x < 2}$. What are the upper bounds, lower bounds, greatest lower bounds and least upper bounds?
Here is what i know.
Definition, consider the poset $A$ and its subset $B$. An element $ain A$ is called an upper bound if $bleq a$ for all $bin B$. An element $ain A$ is called an lower bound if $aleq b$ for all $bin B$.
So my first go was, that $1$ is the greatest lower bound with lower bounds $1leq x < 2$ in $P$, and there exists no upperbound since $2$ is not smaller than $2$. And since there is no upper bound, there cannot be a least upper bound.
However, this when I ONLY consider the numbers in $P$ ($P$ is the universe). If i would consider $P$ to be a subset, that is $Psubseteq (mathbb{R}, leq)$, then I would guess the least upper bound is $2$, and all upper bounds are $xin[2,infty)$. Same for lower bounds, $1$ is the greatest lower bound and $xin(-infty, 1]$ are all the lower bounds.
I think the former is totally wrong, since subset definition is not used, and the latter is partially wrong since $2$ is not contained in $P$, but it is included in my upper bounds. However, I am not sure whether or not to consider $P$ to be a subset of anything else in order to give an answer. Are there two ways to look at this problem?
In desperate need for some enlightening.
discrete-mathematics
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
$endgroup$
I am following a course in discrete mathemathics and I am having great dificulty in understanding the lower, upper bounds and its lub's and glb's. This keeps me from fully understanding everything else.
Consider the set $P = {xmid xinmathbb{R} , 1 leq x < 2}$. What are the upper bounds, lower bounds, greatest lower bounds and least upper bounds?
Here is what i know.
Definition, consider the poset $A$ and its subset $B$. An element $ain A$ is called an upper bound if $bleq a$ for all $bin B$. An element $ain A$ is called an lower bound if $aleq b$ for all $bin B$.
So my first go was, that $1$ is the greatest lower bound with lower bounds $1leq x < 2$ in $P$, and there exists no upperbound since $2$ is not smaller than $2$. And since there is no upper bound, there cannot be a least upper bound.
However, this when I ONLY consider the numbers in $P$ ($P$ is the universe). If i would consider $P$ to be a subset, that is $Psubseteq (mathbb{R}, leq)$, then I would guess the least upper bound is $2$, and all upper bounds are $xin[2,infty)$. Same for lower bounds, $1$ is the greatest lower bound and $xin(-infty, 1]$ are all the lower bounds.
I think the former is totally wrong, since subset definition is not used, and the latter is partially wrong since $2$ is not contained in $P$, but it is included in my upper bounds. However, I am not sure whether or not to consider $P$ to be a subset of anything else in order to give an answer. Are there two ways to look at this problem?
In desperate need for some enlightening.
discrete-mathematics
discrete-mathematics
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
asked Jan 8 at 16:03
CapitalMattersCapitalMatters
83
83
$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32
add a comment |
$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32
$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32
add a comment |
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$begingroup$
You are right, P is presumed to be a subset of R.
$endgroup$
– William Elliot
Jan 8 at 22:15
$begingroup$
@WilliamElliot Thanks, i understand now.
$endgroup$
– CapitalMatters
Jan 9 at 16:32