Analyze the following relation R defined on the set of all people.

Question

 Analyze the following relation R defined on the set of all people. Determine whether R is an equivalence relation. Specify which of the relations are reflexive, symmetric, antisymmetric and/or transitive. Justify your answer using examples. a) p1 p2 for people if p1 and p2 have a grandparent in common b) p1 is taller than p2 

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Answer 1

Remember a relation is an equivalence relation if it is reflexive, symmetric, and transitive. 

Let's start with a.

For it to be reflexive it would have to be that a person has a grandparent in common with themselves.  This is trivially true.  Likewise if two people have a grandparent in common, then it doesnt matter which goes for in the relation (ie if p1 p2 or p2 p1), both are true.  So it is also symmetric.  Suppose now that p1Rp2 and p2Rp3, that is p1 and p2 share a grandparent and p2 and p3 share a grandparent.  Can we come up with an example where p1 and p3 don't share a grandparent? It is possible(I can spell it out but you can probably figure out how for yourself), therefore the relation is not transitive.   This also means that the relation is not an equivalence relation since it is not transitive

 

For b, it is clear that a person cannot be taller than themselves so it is not reflexive. Similarly it is obvious that if p1 is taller than p2 then p2 cannot be taller than p1.  Therefore it is antisymmetric.  However if p1 is taller than p2 and p2 is taller than p3, clearly p1 is taller than p3.  This shows it is transitive.  However since it is not reflexive and symmetric it is not an equivalence relation.  

 

Hope this helps