The following set has been given: $A = \{1,2,3\}$, and the following relation on $A$ has been given: $S = \{(1,1),(2,1),(1,2),(2,2),(3,3)\}$. The answer says this is a valid equivalence relation. I can see how it is symmetric and reflexive, but I can't see how and why it is transistive. What am I not understanding?
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It obviously has equivalence classes ${1,2}$ and ${3}$. Two elements are related iff they are in the same equivalence class. – almagest May 17 '16 at 18:22
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Check that $\;(a,b)\in S,\,(b,c)\in S\implies (a,c)\in S\;$ for all the possible instances. For example
$$(1,2)\in S,\,(2,1)\in S\implies (1,1)\in S\;\;\color{green}\checkmark$$
There is only one more instance to check...
Did
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DonAntonio
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Can you find two pairs $\langle a,b\rangle,\langle b,c\rangle\in S$ such that $\langle a,c\rangle\notin S$?
If not then you can conclude that the relation is transitive.
drhab
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