कक्षा 9–12Foundationsनियोजित
Countability and Cantor's diagonal argument
Comparing infinite sets by bijections: ℕ, ℤ and ℚ are countable, every set is strictly smaller than its power set, and Cantor's diagonal argument shows that the real numbers are uncountable.
math.found.countability
इस विषय का पाठ अभी लिखा नहीं गया है। नीचे दी गई रूपरेखा पाठ्यक्रम-मानचित्र का हिस्सा है और इसकी समीक्षा नहीं हुई है।
इस पर आधारित
14 अनिवार्य पूर्वापेक्षाएँ (बिल्कुल नीचे तक)
आप सीखेंगे
- Show that ℤ and ℚ are countable by constructing explicit enumerations or bijections.
- Define the power set P(A) and prove Cantor's theorem that no function A → P(A) is surjective.
- Reproduce Cantor's diagonal argument that [0, 1) is uncountable, handling numbers with two decimal expansions.
परिभाषा
- Sets A and B have the same cardinality if there is a bijection A → B.
- A set is countable if it is finite or has the same cardinality as ℕ; otherwise it is uncountable.
- The power set P(A) is the set of all subsets of A.
प्रमेय
- ℤ, ℚ and ℕ × ℕ are countable.
- A countable union of countable sets is countable (the general statement uses the axiom of countable choice).
- Cantor's theorem: for every set A there is no surjection from A onto P(A).
- ℝ is uncountable; in fact ℝ and P(ℕ) have the same cardinality.
आम भ्रांतियाँ
- Thinking a proper subset of an infinite set must be smaller: ℕ and the even numbers have the same cardinality.
- Objecting that the diagonal number 'can just be added to the list'; the argument shows that every list misses some number.
- Forgetting that 0.4999… = 0.5, which a careful diagonal argument must avoid (for example by never changing a digit to 0 or 9).
आगे ले जाता है
वैकल्पिक खोजें
- Forcing and independence Is there a set strictly larger than ℕ but smaller than ℝ? This is the continuum hypothesis; Gödel (1940) and Cohen (1963) showed that, if ZFC is consistent, it can be neither proved nor refuted in ZFC.विस्मय पथ →
- Gödel's incompleteness theorems A diagonal idea reappears in Gödel's proof through a self-referential sentence; the resemblance is genuine in spirit, but Gödel's diagonal lemma is a different construction.
- ← Counting
- ← Sets
- ← Set-builder notation, quantifiers and infinite sets
- ← Rational numbers
अधिगम पथ
आगे पढ़ने के लिए
- Paul R. Halmos, Naive Set Theory (1960)
- Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK (1998)