Let X be the set {1,2,3,4} and let R={(1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4) }. Show that R is an equaivalence relation and write down its equvalence classes.
I don't really understand what does equivalence relation mean.
Reflexive: $\displaystyle aRa ~ \forall a \in R$
Transitive: $\displaystyle \text{If} ~ aRb ~ \text{and} ~ bRc \implies aRc ~ \forall a,b,c \in R$
Symmetric: $\displaystyle aRb \implies bRa ~ \forall a,b \in R$
For example, this relation is reflexive because 1R1, 2R2, 3R3, 4R4. (As can be seen by noting that $\displaystyle \{ (1, 1), (2, 2), (3, 3), (4, 4) \} \in R$.)
You can show that it is symmetric because 1R2 and 2R1, etc. ($\displaystyle \{ (1, 2), (2, 1) \} \in R$) Note that we need not require that 3R1 because (1, 3) is not in the set R.
I'll leave transitivity to you (that's the long one and I'm lazy. )
-Dan