· 3 min read

Learn Set Theory Basics: 36 Flashcards from Unions and Power Sets to Russell's Paradox

Learn set theory basics: union, intersection, symmetric difference, power sets, Cantor's theorem, Russell's paradox and ZFC, with worked examples and tips.

Overlapping translucent Venn diagram circles with glowing dots among floating flashcards

Studying. But it's also a video game. 🎮

Flashcards, quizzes and games that make you actually remember.

Start playing — it's free

Posted by

Set theory is the language the rest of mathematics is written in: functions, relations, probability and logic all reduce to sets. The Flip It deck Set Theory Basics (code XV7279, 36 cards) splits cleanly into two halves: the operations you will actually compute with, and the history of how the foundations nearly collapsed.

Study this deck: Set Theory Basics

1 / 10
Question

Set theory

Knew itLearning

Tap the card for the answer · swipe right if you knew it, left if not

What you'll memorize

Operations and notation: membership (o ∈ A), roster and set-builder description, subset (⊆), proper subset, union (∪), intersection (∩), set difference (∖), complement (Aᶜ), symmetric difference (△), Cartesian product (×), empty set (∅), power set P(A), Venn diagrams.

Foundations: Cantor's uncountability of the reals, Cantor's theorem, aleph (ℵ) for cardinals and omega (ω) for ordinals, cardinality by one-to-one correspondence, Russell's paradox, Frege's Basic Law V (unrestricted comprehension), naive set theory, ZFC, pure sets, the von Neumann universe and its ranks, Paul Cohen's forcing.

Sample cards

FrontBack
Union of {1, 2, 3} and {2, 3, 4}{1, 2, 3, 4}
Intersection of {1, 2, 3} and {2, 3, 4}{2, 3}
Set difference {1, 2, 3} ∖ {2, 3, 4}{1}
Symmetric difference of {1, 2, 3} and {2, 3, 4}{1, 4}
Cartesian product of {1, 2} and {red, white}{(1, red), (1, white), (2, red), (2, white)}
Power set of {1, 2}{ { }, {1}, {2}, {1, 2} }
Is a set a subset of itself?Yes, though not a proper subset
Cantor's theoremThe power set of A is strictly larger than A, even when A is infinite
Rank of the empty set in the von Neumann hierarchyRank 0

The strength of this deck is that every operation comes with a worked example on the same two sets, {1, 2, 3} and {2, 3, 4}. Keep that pair in your head and you can re-derive any definition.

Two tips specific to sets

Map each operation to a logic connective. Union is OR, intersection is AND, complement is NOT, symmetric difference is XOR, subset is implication (x ∈ A → x ∈ B). If you already know truth tables, you get the whole first half of the deck almost for free, and a Venn diagram is just a truth table drawn as regions.

Add the facts the deck names but never states. A few items that exam questions test are missing:

  • |P(A)| = 2ⁿ. Card 20 lists four subsets of a two-element set but never gives the formula, which is also what makes Cantor's theorem intuitive.
  • ∅ ⊆ A for every set A, the classic trick question.
  • De Morgan's laws: (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ and (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ. Not a single card covers them.
  • Countable. Card 21 says the reals are uncountable without defining countable (can be listed in a sequence, like ℕ, ℤ and ℚ; size ℵ₀).
  • What forcing proved. Card 35 describes Cohen's method but not its result: in 1963 it showed the continuum hypothesis cannot be proved from ZFC.

Also note that card 13 writes the complement as Ac. It is the superscript Aᶜ that lost its formatting.

Order of study

Do cards 1-20 first, with the logic mapping above, until the example sets are automatic. Then treat 21-36 as a story in order: Cantor's infinities → Frege's axiom → Russell's paradox → ZFC as the repair → von Neumann's hierarchy → forcing.

Related reading: propositional logic and truth tables and the broader discrete mathematics study guide.

Open XV7279 — Set Theory Basics in the library on Flip It.

Ready to learn faster? 🚀

Join FLIP IT for free and turn studying into a game.

Create my free account