Must pigeonholes partition?

Question

My course notes (Mathematics BSc, third-year module in combinatorics, unpublished) have a proposition,

Let $A$ be a finite set, with $|A| = m$ say. Suppose we have a list of subsets $P_1,\dots,P_n$ (called pigeonholes) such that every element of $A$ lies in one of these subsets. Suppose that $m > n$ (so the number of elements is greater than the number of pigeonholes). Then there exists $i$ such that $|P_i| > 1$.

But must every element of $A$ lie in exactly one of these subsets, or at least one? I think the proposition holds in either case.

For instance, let $A=\{1,2,3\}$ and let $P_{1,1}=\{1\},P_{1,2}=\{2,3\},P_{2,1}=\{1\},P_{2,2}=\{1,2,3\}$. Then $P_{1,1},P_{1,2}$ and $P_{2,1},P_{2,2}$ both satisfy "every element of $A$ lies in at least one of these subsets", but only $P_{1,1},P_{1,2}$ satisfies "every element of $A$ lies in exactly one of these subsets". Are they both 'pigeonholings' of $A$?

Answer 1

According to the definition, both are "pigeonholdings" of $A$ as there is no constraint on number of times the element can appear.

As you mentioned, the property is satisfied in both cases.

Answer 2

"every element of lies in one of these subsets" says in at least one. And yes both of your examples are Pigeonholes.