Given partial orders (A, R) and (B, S), a function f : A → B is called orderpreserving if for all $x, y ∈ A, xRy ⇒ f(x)Sf(y)$. How many such order-preserving functions are for each of the following, where $R, S$ both denote ≤ (the usual "less than or equal to" relation)?
a) A = {1, 2, 3, 4}, B = {1, 2};
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I'm not sure I understand this question. So any function where f(x) ≤ f(y) and f(x), f(y) ∈ {1,2} should be counted? Well, that number is infinite: f(x) = x/x, 2x/2x, 3x/3x... etc etc