# Is “$A_i=A_j$” in the definition of a partition correct?

"Definition 5 Let X be a nonempty set. By a partician P of X we mean a set of nonempty subsets of X such that:

(a) If A, B$\in$P and A$\neq$B, then A$\bigcap$B=$\emptyset$ (b) $\bigcup \limits_{C \in P}C$ = X"

Source: Set Theory by You-Feng. Lin and Shwu-Yeng T. Lin

"3.7 Definition Let A be a set; by a partition of A we mean a family {$A_i$} $_{i \in I}$ of nonempty subsets of A with the following properties:

P1. $\forall i,\space j\in I, A_i \bigcap A_j = \emptyset \space or \space A_i=A_j$.

P2. A = $\bigcup \limits_{i \in I}A_i$"

Source: Set Theory Charles C. Pinter.

Why the definition in the first book defines the partition "(a) A, B$\in$P and A$\neq$B then A$\bigcap$B=$\emptyset$", while the second defines the partition as " P1. $\forall i,\space j\in I, A_i \bigcap A_j = \emptyset \space or \space A_i=A_j$" even if they are defining about the same concept, a partition?

If "$A\neq B$ then $A$" is in the former book, shouldn't "$A_i=A_j$" in the second book be "$A_i \neq A_j$" since they are the definins of a partition? When I compare the second book by Charles C. Pinter with the book by Susann Epp, I think "$A_i=A_j$" is not correct, since $A_i$, $A_j$ should be disjoints.

Source: Discrete Mathematics with Applications by Susanna Epp

It's correct.

In the definition they don't a priori assume that $A_i\ne A_j$, that's why they have to add the "or $A_i=A_j$ part.

Actually an implication is a special kind of logical disjunction. $\phi\Rightarrow \psi$ is the same as $\neg\phi\lor\psi$ ($\psi$ is true or $\phi$ is false).

That's what's in play here when describing the mutual disjointness. So instead of saying that if $A_i\ne A_j$ then $A_i\cap A_j=\emptyset$ they say $A_i\cap A_j=\emptyset$ or $A_i=A_j$.

• I see. I looked the second definition again, and noticed "a family {$A_i$} $_{i \in I}$ of nonempty subsets of A" So in P1, "$A_i \bigcap A_j = \emptyset or A_i=A_j$", it's immediately deduced that $A_i=A_j \neq \emptyset$, and if there's no "$or A_i=A_j$" in the definition, it can mean $A_i \bigcap A_j=A_i = \emptyset$, but it shouldn't mean that. In addition, it's not "$A_i \bigcap A_j = \emptyset AND A_i=A_j$", that's what's missed in the second book. Wow, now I think I understand it completely. – buzzee Jan 22 '16 at 11:36

The second definition speaks of a family $\{A_i\}_{i\in I}$ that satisfies certain conditions. If it does then the set $\{A_i\mid i\in I\}$ will satisfy the conditions mentioned in the first and third definition.

So actually the concepts that are described are not fully the same (set against family).

Note e.g that family $\{A_i\}_{i\in\{1,2\}}$ where $A_1=A_2=X$ is a partition of set $X$ according to the second definition, and that set $\{A_1,A_2\}=\{X,X\}=\{X\}$ is a partition of set $X$ according to the first and third definition.

Personally I prefer and am used to the first definition (so no indices).

It's fine like that. Because if $i=j$ then $A_i\cap A_j=\varnothing$ exactly when $A_i=\varnothing$, which is impossible in the definition of a partition.

So we allow equality to hold, so we don't have to say $\forall i\neq j\in I, A_i\cap A_j=\varnothing$. Which is slightly more awkward to understand.