Set operations: turn membership into an argument

A set is determined by its members. Repeating an entry or changing the written order does not change it: the collections written as $\{2,2,5\}$ and $\{5,2\}$ are equal. Membership asks whether a particular object is in a set. Containment asks whether every member of one set is also a member of another. An element can itself be a set, so the two questions must remain distinct even when both sides use braces.

The empty set contains no elements and is a subset of every set: a counterexample to containment would need an element of the empty set, and none exists. However, the empty set is not automatically an element of every set. For example, the singleton containing the empty set has one element, whereas the empty set has zero. Keeping this difference explicit prevents mistakes when later counting collections of subsets or constructing partitions.

An object belongs to a union when it belongs to at least one operand, and to an intersection when it belongs to both. Set difference keeps members of the first set that fail membership in the second. A complement needs a declared universe: the complement of the even integers among integers differs from their complement among real numbers. In a database analogy, the universe is the table being filtered, not every conceivable object.

To prove an identity, pick an arbitrary object and translate each operation into and, or, and not. For De Morgan's law, failing to belong to a union means satisfying neither membership condition, so it means belonging to both complements. This argument establishes equality for all possible objects at once. Testing three sample objects can expose a false identity, but it cannot replace the arbitrary-object argument establishing a true one.

Distribution describes how two kinds of filtering interact. Membership in an intersection with a union means satisfying the first condition and at least one of the other two; this can be rearranged into a union of two intersections. A Venn diagram is useful for inventing the claim, while a membership proof explains why it survives even for infinite sets or sets with no meaningful geometric shape.

A power set is the set of all subsets of a given set, including the empty subset and the whole set. It is a new level of collection, not a synonym for the original set. For two distinct objects there are four subsets, because each object independently receives a yes or no membership choice. Later counting arguments will turn this decision process into a general formula, and diagonal arguments will show why power sets matter even beyond finite counting.