Difficult# Inverse Image, Preimage

ABSALG-NTGEEQ

Let $S$ and $T$ be two (non-empty) sets, and let $f:S\rightarrow T$ be a function with domain $S$. Let $B\subseteq T$.

The __ inverse image__ (or

$$f^{-1}(B)=\{s\in S\mid f(s)\in B\}$$

To every $A\subseteq S$, we associate the function $f=f_A$ given by $f_A(a)=1$, for $a\in A$, and $f_A(a)=0$ for $a\not\in A$.

Consider the following lines from The Owl and the Pussy-Cat, by Edward Lear:

*The Owl and the Pussy-cat went to sea; In a beautiful pea-green boat;They took some honey, and plenty of money;Wrapped up in a five-pound note;The Owl looked up to the stars above; And sang to a small guitar...*

Let $S=\{$Owl, Pussy-Cat, pea-green boat, honey, plenty of money, five-pound note, guitar$\}$.

Consider the ordering $\preceq$ on $S$ given by the inclusion relation determined colloquially by the above verse.

For example, honey $\prec$ five-pound note, as honey is wrapped up in the five-pound note, and Owl $\prec$ pea-green boat, as the Owl is in the pea-green boat.

(We can make $\preceq$ into a partial ordering by supposing $s\preceq s$ for all $s\in S$).

We call $t\in S$ __the least upper (resp. greatest lower) bound of a subset $A$ of $S$__ if:

i.$a\preceq t$ (resp. $t\preceq a$) for all $a\in A$

ii.if there is a $u\in S$ with $a\preceq u$ (resp. $u\preceq a$) for all $a\in A$, then $t\preceq u$ (resp. $u\preceq t$).

Which of the following is **TRUE**?

We use the above notations for the functions $f_A:S\rightarrow \{0,1\}$, $A\subseteq S$.