Proper subset
Elementary
Think of two sets, $A$ and $B$, where everything in $A$ is also in $B$. The set $A$ is a proper subset of $B$ if $B$ has at least one thing that $A$ does not.
High school
Consider two sets $A$ and $B$ such that $A \subset B$. The set $A$ is a proper subset of $B$ if $A \neq B$, so $B$ has at least one element that is not in $A$.
Examples
Examples
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$\emptyset$ and $\setdef{1, 2}$
Elementary
$\emptyset$ is a proper subset of $\setdef{1, 2}$ because the empty set is a subset and $\emptyset \neq \setdef{1, 2}$.
Non-examples
Things that may be mistaken for examples.
- $1$ and $\setdef{1, 2}$ Elementary
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$\setdef{1, 2}$ and $\setdef{1, 2}$
Elementary
$\setdef{1, 2}$ is not a proper subset of itself, because a proper subset must leave out some element.
- $\setdef{1}$ and $\setdef{\setdef{1}, 2}$ Elementary