To find subbasis for a topology on X. Consider the discrete topology $\tau$ on $X:= \{ a,b,c, d,e \}$. Find subbasis for $\tau$ which does not contain any singleton sets.

The definition of subbasis is as follows: 


  Definition: A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union is $X$.


So let $S$ be equal to the  collection of $\{a,b\}$, $\{c,d\}$ and  $\{d,e\}$.  

Clearly union of these three elements is $X$.  

So should be $S$ - as defined - be taken as subbasis? Please check the answer I posted in comment.
Hint: You can write $\{a\}$ as $\{a,b\}\cap\{a,c\}$. Do the same with each of the elements of $X$.
The correct singleton sets should be $\{(a,b),(b,c),(c,d),(d,e),(a,e)\}$