Elementary Set Theory ~ Partitions

Question

I tried searching for a related thread to this, so please don't roast me too hard if one already exists.

Anyways, if I have a set $A = \{a, b, c\}$ then $\{a, b, c\}$ would not be considered a partition of $A$ while $\{\{a, b, c\}\}$ would be a partition, why is this? Also, would $\{a, b, c, ∅\}$ be considered a partition of $A$?

Thanks!!

Answer 1

A partition of a set $A$ is a set $S$ of subsets of $A$ which satisfy:

1. $\varnothing\notin S$.
2. If $X,Y\in S$ then either $X=Y$ or $X\cap Y=\varnothing$.
3. For every $a\in A$ there is some $X\in S$ such that $a\in X$.

The set $\{a,b,c\}$ is a subset of $A$ but not necessarily a set of subsets of $A$. Likewise, $\{a,b,c,\varnothing\}$ is generally not even a subset of $A$ to begin with.

Answer 2

A partition is a set of sets whose union is the universal set and intersection is the null set.
In your case, $\{a,b,c\}$ is not a partition because, its elements $a ,b, c$ are not sets. $\{\{a,b,c\}\}$ is a partition because its only element $\{a,b,c\}$ is the set $A$. Other examples of partitions: $\{\{a\},\{b\},\{c \}\} \{\{a\} ,\{b,c\}\}$ etc...
No. $\{a,b,c,\varnothing\}$ is not a partition.