{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Cardinal Arithmetic

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

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

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.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern