The proposition given is:- every equivalence relation induces a partition of the underlying set, the parts of the partition being the equivalence classes, i.e. the equivalence classes are pair-wise disjoint subsets whose union is the whole set.
Take a set $A = \{a,b,c,d,e\}$ and a relation on $A$, $R = \{(a,a),(b,b),(c,c),(a,c),(c,a)\}$. $R$ is equivalent because it's reflexive, symmetric and transitive. $R$ will only give the equivalent classes $[a]$, $[b]$ and $[c]$ which are $\{a,c\}$ and $\{b\}$. the union of these equivalent classes is $\{a,b,c\}$ which is not equal to $A$ so how does the above proposition work?