Definition:Comparable Sets
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Definition
Comparable in Size
Let $S$ and $T$ be sets.
Then $S$ and $T$ are comparable (in size) if and only if either:
- $S$ can be put into one-to-one correspondence with a subset of $T$
or:
- $T$ can be put into one-to-one correspondence with a subset of $S$
or both.
That is, if either $S$ is smaller than $T$ or $T$ is smaller than $S$.
Comparable by Subset Ordering
Let $S$ and $T$ be sets.
Then $S$ and $T$ are comparable (with respect to the subset ordering) if and only if either:
- $S \subseteq T$
or:
- $T \subseteq S$
or both.
Also see
- Definition:Comparable Elements (of a general relation)
- Results about comparable sets can be found here.