Give me an example of a relation.

Question

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?

Answer 1

(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.

Answer 2

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.