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In the context of donor-recipient compatibility, human blood types are typically divided into 8 classifications: O-, O+, A-, A+, B-, B+, AB-, and AB+. Unfortunately, using this notation makes it difficult to remember which types can donate to each to other. Can A- donate to B+? Who knows. Unless someone has an interest in medicine, it is unlikely that they know much about donor-recipient compatibility beyond perhaps that O- is the universal donor. A typical blood type compatibility table (as shown on the right) is unnecessarily complex and obscures the natural underlying structure. Who would want to commit that monstrosity to memory?
Robbie Angarone and I would like to introduce the XYZ system for organizing and displaying blood type compatibility. The 8 blood type classifications in this system are the sets ∅, {X}, {Y}, {Z}, {X,Y}, {X,Z}, {Y,Z}, and {X,Y,Z}. Under this notation, we can describe the valid donor-recipient pairs with a very simple rule:
A can donate to B if and only if A ⊆ B
The structure described here is a partially-ordered set, or poset. In particular, it is the boolean lattice of the set {X, Y, Z} and its Hasse diagram forms a three-dimensional cube, as shown below. By mapping X = A, Y = B, and Z = +, we can recover the original blood type classifications while preserving the pleasing and intuitive symmetric structure.
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