Set, Subset, Equality Of Sets

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Definition (Set, Subset, Equality Of Sets)

A set $S$ is a collection of elements that have been grouped together.

• We use brackets $\{$ and $\}$ to enclose elements of sets.
• We'll write $x\in S$ to say that $x$ is an element of $S$ or $x$ is in $S$. Similarly, we'll write $x\notin S$ to say that $x$ is not in $S$.
• We say that a set $B$ is a subset of the set $S$, and we write $B\subseteq S$, if every element in $B$ is also an element of $S$. We also read $A\subseteq B$ as "$A$ is contained in $B$." We'll often write $B\supseteq A$ instead of $A\subseteq B$, and read $B\supseteq A$ as either "$B$ is a super set of $A$" or "$B$ contains $A$."
• We say that $B$ is a proper subset of $S$ if $B\subseteq S$ but there is an element of $S$ that is not in $B$.
• We say that two sets $A$ and $B$ are equal if $A\subseteq B$ and $B\subseteq A$.