Puzzle 75 is about things like this "a dozen eggs costs more than a stick of butter". We have a set \\(\mathbb{R}\\) whose elements are _amounts of money_, ordered in the usual way. We have a set \\(X\\) whose elements are _things you can buy in the grocery store_. And we have a function \\(f: X \to \mathbb{R}\\) mapping each thing you can buy in the grocery store to its price. Say

\[ f(\text{a dozen eggs}) = 3.50 \]

and

\[ f(\text{a stick of butter}) = 0.75 \]

Then we say

\[ \text{a stick of butter} \le_X \text{a dozen eggs} \]

because

\[ 0.75 \le_{\mathbb{R}} 3.50 .\]

This is just a way of saying that a stick of butter is cheaper than a dozen eggs. It makes perfect sense. Please, someone do these puzzles!

By the way, Brandon passed his thesis defense, and all my students are happy. =D> =D> =D> =D> =D> =D>

\[ f(\text{a dozen eggs}) = 3.50 \]

and

\[ f(\text{a stick of butter}) = 0.75 \]

Then we say

\[ \text{a stick of butter} \le_X \text{a dozen eggs} \]

because

\[ 0.75 \le_{\mathbb{R}} 3.50 .\]

This is just a way of saying that a stick of butter is cheaper than a dozen eggs. It makes perfect sense. Please, someone do these puzzles!

By the way, Brandon passed his thesis defense, and all my students are happy. =D> =D> =D> =D> =D> =D>