What is the difference between Partition and Division?

While reading graph theory, I came across different definitions where they use partitions and divisions, I was wondering, are these terms same or different?

Can anyone explain me their difference in Set Theory?

I know this is a simple question but anyone hardly talk about it but they always make here a mistake.

• I'm having doubts about the term "Graph Division"... may be, you mean "Graph Subdivision"? Apr 28 at 3:48
• Please give a source, there are different terminologies and the meaning for sets might not be the meaning for graphs. Apr 29 at 12:13

I will explain this with an example :

Consider a set $$S$$ which contains a collections of sets with a constant k such that: $$S = \{ \{S_1\}, \{S_2\}, \{S_3\},\{S_4\},..........\{S_k\} \}$$

In Partition of Set S :

Then we can form the set again by

$$S=S_1\cup S_2\cup S_3\cup S_4...........\cup S_k$$

$$where\ S_i\cap S_j = \phi$$

$$and\ i\neq j\ and\ i,j\in \mathcal{N}$$

In Division of Set S :

We just have this

$$S=S_1\cup S_2\cup S_3\cup S_4...........\cup S_k$$

• The definitions of $S$ (or rather the $S_i$) does not match its (their) use! Furthermore, do you hae any source for this terminology (I mean division; partition is fine)? Apr 29 at 12:05
• Division is not a standard concept in combinatorics. There is no reason to assume that division has this meaning. May 3 at 7:16