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The quotient set: \(\mathbb X / R = \{ [a] : a \in \mathbb X \}\)
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+Posets
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+Partial orders
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+- \(\mathbb X\) is a set, \(R \subseteq \mathbb X \times \mathbb X\)
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+Def: \(R\) is a partial order on \(\mathbb X\) iff \(R\) is:
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+- Reflexive
+- Antisymmetric
+- Transitive
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Def: \(m \in \mathbb X\) is said to be:
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+- maximal element in \((\mathbb X, \preccurlyeq)\) iff \((\forall a \in \mathbb X) m \preccurlyeq a \implies m = a\)
+- largest iff \((\forall a \in \mathbb X) (a \preccurlyeq m)\)
+- minimal iff \((\forall a \in \mathbb X) (a \preccurlyeq m \implies a = m)\)
+- smallest iff \((\forall a \in \mathbb X) (m \preccurlyeq a)\)
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Def: A partial order \(R\) on \(\mathbb X\) is said to be "total" iff \((\forall x, y \in \mathbb X) (x R y \lor y R x)\)
Def: A subset \(B\) of \(\mathbb X\) is called a chain "chain" iff \(B\) is totally ordered by \(R\)
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+- \(C(\mathbb X)\) - the set of all chains in \((\mathbb X, R)\)
+- A chain \(D\) in \((\mathbb X, R)\) is called a maximal chain iff \(D\) is a maximal element in \((C(\mathbb X), R)\)
+- \(K \subseteq \mathbb X\) is called an antichain in \((\mathbb X, R)\) iff \((\forall p, q \in K) (p R q \lor q R p \implies p = q)\)
+- Def: \(R\) is a partial order on \(\mathbb X\), \(R\) is called a well order iff \(R\) is a total order on \(X\) and every nonempty subset \(A\) of \(\mathbb X\) has the smallest element
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+Induction
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+- If \(\phi\) is a propositional function defined on \(\mathbb N\), if:
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+- \(\phi(1)\)
+- \((\forall n \geq 1) (\phi(n) \implies \phi(n+1)\)
+- \((\forall k \geq 1) \phi(k)\)
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