Determine whether the relation R on the set of all people is reflexive, symmetric, anti- symmetric, and/or transitive, where (a, b) ϵ R if and only if a is taller than b.
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Recall reflexivity means that our relation is true between an element and itself; equality of integers is reflexive because we know for any integer \(n\) that \(n=n\) (this is an axiom that the most basic math such as elementary algebra necessitates). Consider our relation, though; if we take any person, can you say he is taller than himself? That seems paradoxical, no?
Yes it does
Consider our next property, symmetry, which says that if an element a is related to b by our relation, then b must be similarly related to a by our relation. If we two people, say Alice and Bob, and we know Alice is strictly taller than Bob, does that mean Bob is strictly taller than Alice? This, too, makes very little sense!