Set Theory

Every combination of two sets, with distinct roles for the first and second positions.

How to partition a union into “only A,” “only B,” and the shared part without losing or double-counting elements.

How to distribute union and intersection across parentheses, and why both identities hold.

Why negation exchanges union and intersection, and how element membership proves both identities.

Why “everything outside a set” is meaningful only after its universe has been fixed.

How to remove the second set’s elements from the first, and why reversing the arguments changes the result.

How to keep only shared elements, and why a family intersection requires membership in every set.

Why membership in any one set is enough for a union, including a whole indexed family.

How a candidate domain and a predicate define a set of qualifying elements.

Counting distinct elements, ignoring duplicates, and distinguishing two kinds of emptiness.

How to check that every element of one set is in another, and why inclusion differs from membership.

How to prove set equality element by element or through two inclusions, and why equal size is not enough.

What membership states, why order and repetition do not change a set, and how an empty set differs from a set containing it.