Equivalence of Definitions of Lower Set
Standard style for equivalence definitions
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Theorem
The following definitions of the concept of Lower Set are equivalent:
Let $\left({S, \preceq}\right)$ be an ordered set.
Let $U \subseteq S$.
Then the following are equivalent:
| \((1)\) | $:$ | \(\displaystyle \forall u \in U: \forall s \in S: s \preceq u \implies s \in U \) | ||||||
| \((2)\) | $:$ | \(\displaystyle U^\preceq \subseteq U \) | ||||||
| \((3)\) | $:$ | \(\displaystyle U^\preceq = U \) |
where $U^\preceq$ is the lower closure of $U$.
Proof
By the Duality Principle, it suffices to prove that:
- $(1^*)$, $(2^*)$ and $(3^*)$ are equivalent
where these are the dual statements of $(1)$, $(2)$ and $(3)$, respectively.
By Dual Pairs, it can be seen that these dual statements are as follows:
| \((1^*)\) | $:$ | \(\displaystyle \forall u \in U: \forall s \in S: u \preceq s \implies s \in U \) | ||||||
| \((2^*)\) | $:$ | \(\displaystyle U^\succeq \subseteq U \) | ||||||
| \((3^*)\) | $:$ | \(\displaystyle U^\succeq = U \) |
Their equivalence is proved on Equivalence of Definitions of Upper Set.
$\blacksquare$