Subset
Elementary
A set $A$ is a subset of a set $B$ if everything in $A$ is also in $B$. For example, the set of golden retrivers is a subset of the set of dogs, and dogs are a subset of all animals.
High school
$A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$.
Examples
Examples
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$\emptyset$ and $\setdef{1, 2}$
Elementary
$\emptyset$ is a subset of $\setdef{1, 2}$ because the empty set is a subset of every set.
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$\setdef{1, 2}$ and $\setdef{1, 2}$
Elementary
$\setdef{1, 2}$ is a subset of itself, as is every set.
Non-examples
Things that may be mistaken for examples.
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$1$ and $\setdef{1, 2}$
Elementary
$1$ is not a subset, because it is not a set of elements of $\setdef{1, 2}$.
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$\setdef{1}$ and $\setdef{\setdef{1}, 2}$
Elementary
$\setdef{1}$ and $\setdef{\setdef{1}, 2}$ is not a subset, because its element $1$ is not an element of $\setdef{\setdef{1}, 2}$.
Properties
- Every set is a subset of itself, and the empty set is a subset of every set.
- $A = B$ if and only if $A \subseteq B$ and $B \subseteq A$.
Notes
The symbol $\subset$ is ambiguous: some authors use it to mean $\subseteq$, and others use it to mean proper subset ($\subsetneq$).