{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Cardinal Arithmetic

Cardinal Arithmetic - Math 100 Cardinal Arithmetic 1 A few...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
Math 100 Cardinal Arithmetic Martin H. Weissman 1. A few countable sets Suppose that A and B are sets. The following are basic definitions of “cardinality”: # A # B if and only if there exists an injective function from A to B . # A = # B if and only if there exists a bijective function from A to B . # A # B if and only if there exists a surjective function from A to B . The following are basic properties of cardinality: # A # B if and only if # B # A . The relations , , = are reflexive and transitive. (The Cantor-Schroeder-Bernstein Theorem): If # A # B and # B # A , then # A = # B . If A and B are sets, then either # A # B or # B # A (or both). In addition to the usual natural numbers, there are various infinite cardinal numbers: 0 = # N , and c = # R . Some intuition about finite sets breaks down with infinite sets. Here is a basic example: Theorem 1. The natural numbers and the integers are sets with the same cardinality. In other words, # Z = # N = 0 . Proof. The “inclusion function” from N to Z is an injective function. Hence # N # Z . Thus 0 # Z . Now, we construct an injective function from Z to N as follows: If z is a positive integer, define f ( z ) = 2 z . If z is a negative integer, define f ( z ) = - 2 z + 1. If z = 0, define f ( z ) = 0.
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern