smoothi963 said:
Determine whether the following relation is an equivalence relation:
for all a,b E N aRb <-> a ends in the same digit in which b ends.
Please help, i know that for it to be an equivalence relation it has to be reflexive, symmetric and transitivity. But i do not know how to check if it's any of those. Thank you
In fact, it is reflexible, because every number \(\displaystyle a \in IN\) has exactly the same last digit as itself (in fact, all the digits are the same)
Also, if \(\displaystyle a R b\), that means that \(\displaystyle a\) has the same last digit as \(\displaystyle b\). This implies in turn that \(\displaystyle b\) has the same last digit as \(\displaystyle a\), and hence \(\displaystyle b R a\).
Finally, assume that \(\displaystyle a R b\), and \(\displaystyle b R c\), this means that \(\displaystyle a\) has the same last digit as \(\displaystyle b\) and \(\displaystyle b\) has the same last digit as \(\displaystyle c\). In other words, all \(\displaystyle a, b,\) and \(\displaystyle c\) have the same last digit, which in particular means that \(\displaystyle a\) has the same last digit as \(\displaystyle c\), or equivalently, \(\displaystyle a R c\).
This means that we have an equivalence relation. This equivalence relation partition the set of natural number into 10 equivalence classes:
\(\displaystyle C_i = \{k: \quad k = i + \displaystyle\sum_{n=1}^{\infty} a_n 10^n, \mbox{ for some sequence } a_n \in \{0,1,2,....9\} \}\)
