A subset describes a relationship between two sets: every element of the first must also belong to the second. Equality is stronger: neither set has any elements missing from the other.

The students in one study group form a subset of the students enrolled in their class, provided every group member is enrolled. The group could even include the whole class.

For sets A and B, A is a subset of B when every element of A also belongs to B. The symbol ⊆ permits equality. To show that inclusion fails, find one element of A that does not belong to B.

• $A=B$ means both $A\subseteq B$ and $B\subseteq A$.
• $A\nsubseteq B$ means that at least one element of A is absent from B.
• $\varnothing\subseteq B$ for every set B: the empty set has no element that could violate the condition.
• $A\subsetneq B$ says $A\subseteq B$ and $A\ne B$: a proper subset, which leaves something out.
• The power set $\mathcal{P}(A)$ is the set of all subsets of $A$, and for a finite $A$ it has $2^{\lvert A\rvert}$ elements.

Every element of $A$ is an element of $B$: that is all $A\subseteq B$ claims, and it leaves $B$ free to hold things $A$ does not. Equality asks for the inclusion in both directions at once.

Both 4 and 9 occur on the right. The extra element 1 does not prevent inclusion.

$4\in\{4,9\}$ and $\{4\}\subseteq\{4,9\}$.

The first statement concerns the number 4. The second concerns the set whose only element is 4.

$\{1,2\}\subseteq\{2,1\}$ and $\{2,1\}\subseteq\{1,2\}$.

Every element on either side also appears on the other side, so these sets are equal.

$\subseteq$ allows the two sets to be equal. $\subsetneq$ does not.

• $A\subseteq B$ means every element of A lies in B; one missing element disproves it.
• $A=B$ is the pair of inclusions $A\subseteq B$ and $B\subseteq A$.

Great work! You now know how to:

• Check an inclusion and disprove one with a single element
• Tell proper inclusion from inclusion
• Count the subsets of a finite set
• Keep $\in$ and $\subseteq$ apart