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# 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 preimage) of $B$ is the set denoted by $f^{-1}(B)$ and given by:

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

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 $f_A:S\rightarrow \{0,1\}$, $A\subseteq S$.

A

The least upper bound of $f_{\{{\rm honey,\,plenty \,of \,money}\}}^{-1}(\{1\})$ is five-pound note.

B

The set $f_{\{{\rm five-pound\, note, \,guitar}\}}^{-1}(\{1\})$ has honey as a lower bound.

C

There is no $s\in f_{\{{\rm five-pound\,note}\}}^{-1}(\{0\})$ such that honey $\preceq s$.

D

The set $f_{\{{\rm pea-green\,boat}\}}^{-1}(\{0\})$ is the empty set $\emptyset$.

E

The set $\{$Owl, five-pound note$\}$ has a lower bound in

$f_{\{{\rm honey,\,plenty \,of \,money,\, five-pound\,note}\}}^{-1}(\{0\})$.