This is a theorem from a book. I'm having a hard time on proving it.
Suppose A is a set,$\mathcal{F}\subseteq \mathscr{P}(A)$, and $\mathcal{F} \neq \emptyset$. Then the least upper bound of $\mathcal{F}$ (in the subset partial order)is $\cup \mathcal{F}$ and the greatest lower bound of $\mathcal{F}$ is $\cap \mathcal{F}$.
I know that to prove that the least upper bound of $\mathcal{F}$ is $\cup \mathcal{F}$, I need to prove that $\cup \mathcal{F}$ is a upper bound and $\cup \mathcal{F}$ is smallest of the set of upper bounds. The problem is I don't understand how to put my two goals in terms of logical symbols. If someone could put them in terms of logical symbols I could possibly do it. So that's what I need.
Thanks in advance.
These statements are essentially the definition of union and intersection. Saying that $\cup \mathcal{F}$ is an upper bound of $\mathcal{F}$ means that for all $B \in \mathcal{F}$, we have $B \subset \cup \mathcal{F}$. This is certainly try. Now to show that $\cup \mathcal{F}$ is the least upper bound, we take any other subset $U$ of $A$ which is an upper bound for $\mathcal{F}$, i.e. with the property that for all $B \in \mathcal{F}$, we have $B \subset \cup \mathcal{F}$, and we must show that $\cup \mathcal{F} \subset U$. Well, suppose we have $x \in \cup \mathcal{F}$. Then by definition, for some $C \in \mathcal{F}$, we have $x \in C$, and since $C \subset U$ by the assumption that $U$ is an upper bound of $\mathcal{F}$, we have $x \in U$, and we are done.
The proof that $\cap \mathcal{F}$ is the greatest lower bound is very similar and you should be able to do it if you understand this proof.