نظریهٔ مجموعهها و عدد اصلی
Set Theory & Cardinality
نظریهٔ مجموعهها و عدد اصلی (کاردینالیتی) مجموعهها.
ریاضیات گسسته کنکور ارشد علوم کامپیوتر
کارتها · 13
Every set is strictly smaller than its power set, X < P(X) , so no surjection (and hence no bijection) X → P(X) exists — which is why P(N) = 2^(ℵ_0) > ℵ_0 .
The set of all functions A→ B has cardinality B ^( A ) , and cardinal-exponent identities collapse most "big-looking" function spaces over a countable domain down to the continuum 2^(ℵ_0) .
Countability survives countable unions, finite-subset collections, and surjective images, because ℵ_0·ℵ_0=ℵ_0 and a surjection f:Btwoheadrightarrow A forces A ≤ B .
For any set X the Hartogs number ℵ(X) is, provably in ZF alone, the least ordinal that does not inject into X — the engine that turns the Axiom of Choice into the Well-Ordering Theorem.
Ordinal sum and product are defined by "concatenating" order types and are associative but not commutative — a finite ordinal is absorbed on the left ( n+ω=ω , 2·ω=ω ) yet sticks out on the right ( ω+n>ω , ω· 2>ω ).
Ordinal exponentiation is defined by a supremum at limit stages, so a finite base raised to ω collapses to ω ( 3^ω=ω ), whereas the same expression read as cardinals gives the continuum ( 3^(ℵ_0)=2^(ℵ_0) ).
Being a member of P(A) means being a subset of A : S ∈ P(A) ⇔ S ⊆ A .
Two ordered sets have the same order type when an order-preserving bijection links them; a well-order (every non-empty subset has a least element) has a unique ordinal as its order type, so being well-ordered and unbounded above does not make a set look like N .
Every nonzero ordinal is either a successor S(β)=β∪ β =β+1 (it has a largest element / immediate predecessor) or a limit ordinal (nonzero, no maximum, equal to the supremum of everything below it); ω is the first limit ordinal, while ω+1 is a successor.
Irrational just means "not a fraction"; algebraic vs. transcendental is a finer split — algebraic numbers are roots of integer polynomials and are closed under arithmetic and radicals, while transcendental numbers like π and e satisfy no such polynomial.
A monotone F:P(A)→ P(A) has a least fixed point, equal to bigcap X : F(X)⊆ X — the intersection of all its pre-fixed points.
Each ZF axiom is a license to form a new set from old ones; knowing which axiom does which job lets you justify every set-building step.
Each finite ordinal is the set of all smaller ordinals, so n= 0,1,…,n-1 ; inside an iterated power set P^k(∅) only finitely many subsets happen to be exactly such ordinals.