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Give me an example of a relation which is:

(i) Reflexive and Symmetric but not Transitive.

(ii)Symmetric and Transitive but not Reflexive.

I'm confused because I think a Ref. and Sym. relation must be Tra. and a Sym. and Tra. relation must be Ref. but I can't prove it. (can't express in rigorous detail, but intutive conjecture seems right.)

Edit 12/2 Example: (ii) If (a,b)$\in$R then (b,a)$\in$R by symmetric-ity and so by transitivity if (a,b)$\in$R and (b,a)$\in$R $\implies$(a,a)$\in$R so doesn't R become reflexive?

RE60K
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2 Answers2

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(i) The 'has eaten lunch with' relation on the set of all people.

(ii) On the set of all people. Two people are related if they were both born in the year 1985.

RE60K
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paw88789
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  • Surely $(ii)$ is reflexive?! – GFauxPas Feb 12 '15 at 17:26
  • @GFauxPas: No because if you weren't born in 1985, then you're not related to yourself (reflexivity is a universal property that must hold for all elements of the set--i.e. $x$ must be related to $x$ for all $x$). – paw88789 Feb 12 '15 at 17:34
  • Oh whoops, I misinterpreted. Thanks. – GFauxPas Feb 12 '15 at 17:37
  • if you aren't born in 1985 then you can't have a relation! – RE60K Feb 12 '15 at 17:49
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    @ADG: Not all elements have to be related to anything. And in fact you need such elements for an example of type (ii)--because for any $x$, if $x$ is related to (some) $y$, then by symmetry $y$ is related to $x$, and then by transitivity $x$ is related to $x$. – paw88789 Feb 12 '15 at 18:16
  • OK ${}{}{}{}{}{}{}{}{}{}$ – RE60K Feb 12 '15 at 18:21
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Using a formal procedure, consider the set $S = \{ 1, 2, 3 \}$ and the relation $$R = \{ (1,1), (2,2), (3,3), (1,2), (2,1), (2,3), (3,2) \}$$ Then $R$ is reflexive and symmetric but not transitive, as $(1,2), (2,3) \in R$, but $(1,3) \not\in R$.

I am sure you can now construct an example for case ii.

Simon S
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