Types of Sets II — Equal Sets, Equivalent Sets, Subsets, Proper Subset and Power Set

Beyond the special sets above, several relationships between sets matter just as much.

Equal sets: two sets $A$ and $B$ are equal, written $A = B$, if they contain exactly the same elements (order and repeated listing never counted). For instance, $\{2, 4, 6\} = \{6, 2, 4\}$.

Equivalent sets: two finite sets $A$ and $B$ are equivalent, written $A \leftrightarrow B$, if they simply have the same number of elements, i.e. $n(A) = n(B)$ — regardless of what those elements actually are. $\{1, 2, 3\}$ and $\{p, q, r\}$ are equivalent (both have 3 elements) but not equal (their elements differ). Every pair of equal sets is automatically equivalent, but equivalent sets need not be equal — the two words sound similar but mean genuinely different things, and this distinction is worth holding onto carefully.

Subset: $A$ is a subset of $B$, written $A \subseteq B$, if every element of $A$ is also an element of $B$:

Two facts follow at once: every set is a subset of itself ($A \subseteq A$), and the empty set is a subset of every set ($\emptyset \subseteq A$ for any $A$).

Proper subset: $A$ is a proper subset of $B$, written $A \subset B$, if $A \subseteq B$ and $A \ne B$ — every element of $A$ lies in $B$, and $B$ has at least one element that $A$ does not. If $A = \{2, 4\}$ and $B = \{2, 4, 6\}$, then $A \subset B$.

Power set: the power set of a set $A$, written $P(A)$, is the set of all possible subsets of $A$ — including $\emptyset$ and $A$ itself. If $A$ has $n$ elements, $P(A)$ has exactly $2^n$ elements:

For example, if $A = \{1, 2\}$, its subsets are $\emptyset, \{1\}, \{2\}, \{1,2\}$ — four subsets in all, matching $2^2 = 4$, so $P(A) = \{\emptyset, \{1\}, \{2\}, \{1,2\}\}$. …