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28 April 2024
| m 08:34 | Subset of Cover is Cover of Subset diffhist +14 Prime.mover talk contribs | ||||
| N 04:32 | Set of Intersections with Superset is Cover diffhist +670 CircuitCraft talk contribs (Created page with "== Theorem == Let $S$ be a set. Let $\CC$ be a cover of $S$. Let $T \supseteq S$ be a superset of $S$. Then: :$\set {C \cap T : C \in \CC}$ is a cover of $S$. == Proof == Let $x \in S$ be arbitrary. By definition of cover, there is some $C \in \CC$ such that: :$x \in C$ By definition of superset: :$x \in T$ Ther...") | ||||
27 April 2024
| N 22:21 | Union of Covers is Cover of Union diffhist +931 CircuitCraft talk contribs (Created page with "== Theorem == Let $\sequence {S_i}_{i \in I}$ be an indexed family of sets. For each $i \in I$, let $\CC_i$ be a cover of $S_i$. Then, $\ds \bigcup_{i \mathop \in I} \CC_i$ is a cover of $\ds \bigcup_{i \mathop \in I} S_i$. == Proof == Let $\ds x \in \bigcup_{i \mathop \in I} S_i$ be arbitrary. By definition of union, there is some $i \in I...") | ||||
| N 19:54 | Cover is Cover of Subset diffhist +515 CircuitCraft talk contribs (Created page with "== Theorem == Let $S$ be a set. Let $\CC$ be a cover of $S$. Let $T \subseteq S$ be a subset of $S$. Then, $\CC$ is a cover of $T$. == Proof == By definition of a cover: :$\ds S \subseteq \bigcup C$ But then, by Subset Relation is Transitive: :$\ds T \subseteq \bigcup C$ Therefore, $C$ is a cover of $T$ by...") | ||||
| 19:54 | Subset of Cover is Cover of Subset diffhist +7 CircuitCraft talk contribs (Proofread, category added.) | ||||