Discrete Mathematics - Vol 2 | 7. Cantor's Theorem by Abraham | Learn Smarter
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7. Cantor's Theorem

This chapter discusses Cantor's theorem and the concept of cardinality in sets. It establishes that the cardinality of any set is strictly less than the cardinality of its power set, and provides various proofs, particularly using the diagonalization argument. The chapter concludes by revealing that there are infinitely many infinities, reflecting the nature of different cardinalities within infinite sets.

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Sections

  • 7

    Cantor's Theorem

    Cantor's Theorem states that the cardinality of any set is strictly less than the cardinality of its power set.

    Learning Practice

  • 7.1

    Introduction To Cardinality

    This section introduces the concept of cardinality and Cantor's theorem, highlighting the relationship between a set and its power set.

    Learning Practice

  • 7.2

    Proof By Contradiction

    Cantor's theorem demonstrates that the cardinality of any set A is strictly less than the cardinality of its power set P(A), using proof by contradiction.

    Learning Practice

  • 7.3

    Construction Of The Subset S

    Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, using a diagonalization argument.

    Learning Practice

  • 7.4

    Implications Of Cantor's Theorem

    Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, establishing the existence of different sizes of infinity.

    Learning Practice

  • 7.5

    Conclusion

    Cantor's theorem demonstrates that the cardinality of any set is strictly less than the cardinality of its power set, including infinite sets.

    Learning Practice

References

ch30 - part B.pdf

Class Notes

Memorization

What we have learnt

  • Cantor's theorem states tha...
  • The proof involves showing ...
  • There are infinitely many d...

Final Test

Revision Tests