2
votes
0answers
23 views

Intuition - Countable iff Surjection iff Injection [Velleman P310 Thm 7.1.5] [duplicate]

Define $I_n = \{1, 2, ..., n \} $. Let $A$ be a nonempty set. TFAE : (i) $A$ is finite (ie: a bijection $h:A\rightarrow I_{N}$ exists) or A is countably infinite (ie: a bijection ...
1
vote
2answers
38 views

Proof strategy - If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]

For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist ...
4
votes
0answers
89 views

Colored Picture for Equivalence Classes, Relations, Partitions, .. [closed]

Origin — A Book of Abstract Algebra — Charles Pinter — p120. I'm trying to sketch a colored picture for the ideas from equivalence classes, equivalence relations, partitions, etc... underneath. ...
3
votes
1answer
160 views

If two sets have the same cardinality, then so do their power sets. Converse can't be answered?

For infinite sets $A, B$, $|A| = |B| \Longrightarrow \require{cancel} \cancel{\Longleftarrow} |P(A)| = |P(B)|$. I recast http://ph.answers.yahoo.com/question/index?qid=20100907061641AAE2Vfq : ...
2
votes
2answers
261 views

Fuzzy Venn diagram regions labeled in ternary

I have a couple of questions about the Venn diagrams object : Words from the binary alphabet with n letters label each region of an order-n Venn diagram. Is there any more profound connection ...
5
votes
1answer
340 views

Intuition about the size of $\aleph_k$ with $k>1$

Assuming CH for simplicity, I know of some more or less intuitive way to think about difference in sizes of $\aleph_0$ and $\aleph_1$. The most straightforward is the distinction of natural/rational ...