define a relation $R$ on $S$?

Question

Let S be the set of humans.

1) Define a relation $R$ on $S$ that is reflexive, symmetric, and transitive but not antisymmetric

2) Define a relation $R$ on $S$ that is symmetric and antisymmetric

Can someone help me understand how to start this? Would we have to create a list of pairs to show the relation?

Answer 1

I don't know how to give a hint to number 1 without giving it away. I'd suggest just trying natural language relationships. Example a R b if b is the mother of a, but is it reflexive? Is a R a? Is a always the mother of a? Of course not. How about if a R b if a and b have the same parents. Is it reflexive? Does a have the same parents as him/herself. Yes, Symmetric, if a has the same parents as b does b have the same parents as a? Yes. Transitive? If a and b have the same parents and b and c have the same parents, do a and c have the same parents?

As for 2). Note . Symmetric means whenever a R b then always b R a. But anti-symmetric says, whenever a R b and b Ra then always a = b. So if a relation is both symmetric and anti-symetric then a R b => b R a = > b = a. So always if a R b then a = b. So no two different elements can be related and only same elements can be related. But a same element doesn't have to be related to itself.

Here are three examples: everybody is related to themselves and no-one else. No body is related to anybody not even themselves. Sam Spade, Barrack Obama, and Kim Kardashian are all related to themselves; nobody else is related to anybody else or to themselves.

Now that I've given you some answers, you need to come up with some completely different answers.

Answer 2

1) Observe that $R$ is demanded to be an equivalence relation. That means that it induces a partition on $S$. Working in opposite direction you can also start with some partition on humans (lots of choices) and then define $R$ by stating that $aRb$ if and only if $a$ and $b$ belong to the same component of the partition. Then automatically $R$ is an equivalence relation.

2) If $R$ is symmetric and antisymmetric then $aRb$ leads firstly to $bRa$ (by symmetry) and leads secondly to $a=b$ (by antisymmetry).

So it is necessary that $$R\subseteq\Delta:=\{\langle a,a\rangle\mid a\in S\}$$ Conversely $R\subseteq\Delta$ is also a sufficient condition for $R$ to be symmetric and antisymmetric.