Analysis
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Introduction to Sets
A set is a collection of objects called elements. The empty set (denoted by $\varnothing$) is the set with no elements. We use the following notation in these notes as a shorthand. Let $S$ be a set. Then:
| Symbol | Meaning |
|---|---|
| $a \in S$ | $a$ is in $S$ |
| $a \notin S$ | $a$ is not in $S$ |
| $\forall$ | For all |
| $\exists$ | There exists |
| $\exists!$ | There exists a unique |
| $:=$ | Define |
| $\implies$ | Implies |
| $\iff$ | If and only if |
In addition, we have the following set relations:
- $A \subseteq B$: $A$ is a subset of $B$ ($x \in A \implies x \in B$)
- $A \subsetneq B$: $A$ is a strict subset of $B$ ($A \subset B$ and $A \neq B$)
- $A = B$: $A$ and $B$ are the same set ($A \subseteq B$ and $B \subseteq A$)