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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?

user332599
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    Which set is S? – Aegis Apr 18 '16 at 18:21
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    What is S defined to be? –  Apr 18 '16 at 18:21
  • Sorry i just found it – user332599 Apr 18 '16 at 18:25
  • So let $S$ be your favorite (preferably finite) set. Just start by making a few ordered pairs of elements of $S$ until you have what you want. – Kevin Long Apr 18 '16 at 18:26
  • What would the ordered pair look like (1,1) , (2,2) and so on? – user332599 Apr 18 '16 at 18:27
  • If your set contains $1$ and $2$, then yes. Just remember what each statement says: Reflexive- everything relates to itself. Symmetric- $x$ relates to $y$ means $y$ relates to $x$. Transitive- $x$ relates to $y$ relates to $z$ means $x$ relates to $z$. Antisymmetric- if $x$ relates to $y$ and vice versa, then $x=y$. – Kevin Long Apr 18 '16 at 18:27
  • so if i was to prove that R on S is reflexive, symmetric , and transitive, it would be like (1,1) (1,2) (2,1) ,(2,2) but not anti symmetric i don't get – user332599 Apr 18 '16 at 18:28
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    First, you should define $S$, say $S={1,2}$. The relation you defined is not antisymmetric because you have two different elements $x$ and $y$ such that $x$ relates to $y$ and $y$ relates to $x$. – Kevin Long Apr 18 '16 at 18:30
  • Non-symmetric and anti-symmetric are not the same thing! Non-symmetric means $\exists x,y;(x R y\land \neg (y R x)).$ Anti-symmetric means $\forall x,y;(x R y\implies \neg (y R x)).$ – DanielWainfleet Apr 18 '16 at 18:55

2 Answers2

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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.

fleablood
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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.

drhab
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