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 $P = \{x\mid x\in\mathbb{R} , 1 \leq x < 2\}$. 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 $a\in A$ is called an upper bound if $b\leq a$ for all $b\in B$. An element $a\in A$ is called an lower bound if $a\leq b$ for all $b\in B$.
So my first go was, that $1$ is the greatest lower bound with lower bounds $1\leq x < 2$ 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 $P\subseteq (\mathbb{R}, \leq)$, then I would guess the least upper bound is $2$, and all upper bounds are $x\in[2,\infty)$. Same for lower bounds, $1$ is the greatest lower bound and $x\in(-\infty, 1]$ 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.