_Products in preorders._

Given a set \\(X\\), a binary operation on \\(X\\) is a function \\(*: X \times X \rightarrow X\\) -- that is, a way of taking two elements of \\(X\\) and returning a third. This question explores how the product unifies many seemingly quite different binary operations commonly used in math.

(a) Consider the category where the objects are natural numbers and where there is a unique morphism from \\(m\\) to \\(n\\) if \\(m\\) divides \\(n\\). Given two numbers, for example 42 and 27, what is their product? What is the name of this binary operation?

(b) Consider the category where the objects are subsets of the set \\(\lbrace a,b,c,d \rbrace\\), and where there is a unique morphism from \\(X\\) to \\(Y\\) if \\(X\\) is a subset of \\(Y\\). Given two subsets, for example \\(\lbrace a,b,c \rbrace\\) and \\(\lbrace b,c,d \rbrace\\), what is their product? What is the name of this binary operation?

(c) Consider the category where the objects are True and False, and where there is a unique morphism from \\(a\\) to \\(b\\) if \\(a\\) implies \\(b\\). Given two objects, for example True and False, what is their product? What is the name of this binary operation?

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