Question

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.

Answer 1

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?

Answer 2

Yes it does

Answer 3

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!

Answer 4

Anti-symmetry states that if we know Alice is taller than Bob for *any* Alice and *any* Bob, then we also know that Bob is NOT taller than Alice. This makes a lot more sense, surely!

Answer 5

yes it does.

Answer 6

Transitivity is another neat property; essentially, if we know Alice is taller than Bob, and Bob is taller than Eve, surely we can infer, then, that Alice is taller than Eve? This is *transitivity*.

Answer 7

... so all in all, we've concluded the relation 'taller than' is neither reflexive nor symmetric but is indeed anti-symmetric and transitive. Neat.

Answer 8

Okay