Equivalent definition of partition in set theory

Question

According to my Prof, the definition of partition in Set-theory is

$S\subseteq P(A) \smallsetminus\{\emptyset\} $ is partition of A if for All $a\in A$ exists $T\in S$ unique such that $a\in T$.

According to How to prove it by Daniel J. Valleman, I see another definition of partition which as following:

Suppose A is a set and $F \subseteq P(A) $ F is called a partition of A if it has the following properties:

1. UF = A.
2. F is pairwise disjoint which means for all $X,Y\in F$ if X is not equal to Y then $X\cap Y = \emptyset$
3. For All $X\in F$ X is not $\emptyset$

My question is how these two definitions is equivalents? I maybe can see a little the the imagination between the second property and uniqueness. and maybe I can understand the third property when we minus an Emptyset.

But I don`t see the imagination between first property to the definition above.

I am looking for clarification regarding these two definitions. Why is it right to use them both in proofs and how can we choose the best definition for our proof.

Answer 1

First note that the following are equivalent:

• For all $a \in A$, there exists $T \in \mathcal{S}$ such that $a \in T$. (This is the 'existence' part of the first definition you have.)
• $\bigcup \mathcal{S} = A$. (This is condition 1 of your second definition.)

To see why, note that $\bigcup \mathcal{S} \subseteq A$ is automatic, and if you flesh out the definitions involved in the statement $A \subseteq \bigcup \mathcal{S}$, you'll see that it's precisely the first statement.

Next note that the following are equivalent:

• For all $T,T' \in \mathcal{S}$, if $a \in T$ and $a \in T'$ for some $a \in A$, then $T=T'$. (This is the 'uniqueness' part of the first definition you have.)
• For all $T,T' \in \mathcal{S}$, if $T \ne T'$, then $T \cap T' = \varnothing$. (This is condition 2 of your second definition.)

To see why, note that the second statement is essentially the contrapositive of the first statement, but with the definition of $T \cap T' \ne \varnothing$ fleshed out as '$a \in T$ and $a \in T'$ for some $a \in A$'.

Finally, the fact that $\mathcal{S} \subseteq \mathcal{P}(A) \setminus \{ \varnothing \}$ is equivalent to the assertion that each $T \in \mathcal{S}$ is non-empty.