Subsets and Set Equality.
Subsets
A set $B$ is called a subset of a set $A$ if every element of $B$ is an element of $A$. This is denoted by $B\subseteq A$.
Examples
\(\begin{aligned} \{3,4,5\}&\subseteq\{1,5,4,2,1,3\}\\ \{3,3,5\}&\subseteq\{3,5\}\\ \{5,3\}&\subseteq\{3,5\} \end{aligned}\)
graph TB
subgraph A
B(B)
end
Venn diagram of $B$ subset $A$.
Therefore, $\forall$ sets $A$, $A\subseteq A$
Furthermore, $\emptyset\subseteq A$ is always true. This is as the empty set is always a subset of any other set including the empty set itself.
Subsets in Python
In programming languages such as python you can save on writing out a function to fund whether a set is a subset of another set. To do this you can use the < symbol in place of the $\subseteq$ symbol:
print n<m
Where n and m are both sets.
Subsets and Bit Vectors Example
Let $S=\langle1,2,3,4,5\rangle,A=\{1,3,5\}$ and $B=\{3,4\}$.
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Is $A\subseteq B$?
$x_a=[1,0,1,0,1]$
$x_b=[0,0,1,1,0]$Therefore $A\nsubseteq B$. As you can see from the aligned bits. Not all the bits present in $x_b$ are present in $x_a$.
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Is the set $C$, represented by $[1,0,0,0,1]$, a subset of the set $D$, represented by $[1,1,0,0,1]$?
$C\subseteq D$ as all bits present in the bit vector of $C$ are also present in the bit vector of $D$.
Equality
As covered before a set $A$ is called equal to a set $B$ if $A\subseteq B$ and $B\subseteq A$. This is denoted by $A=B$.
This is to say that if two sets are subsets of each other then they are equal.
Confirming Equality
Let $S=\langle1,2,3,4,5\rangle,A=\{1,3,5\}$ and $B=\{3,4\}$.
Is $A=B$?
$x_a=[1,0,1,0,1]$
$x_b=[0,0,1,1,0]$
Therefore $A\neq B$ as the bit vectors do not match.