Reference no: EM13990623
1. Prove that for any countably infinite set S, there is a 1-1 function from N onto S. Hint: Let h be a function from N onto S. Define f using h and recursion. The problem is to show that some form of recursion actually applies (1.3.2).
2. To understand the situation in the equivalence theorem: if, in the notation of 1.4.1, f (x ) = y and g(v) = u, say x is an ancestor of y and v an ancestor of u. Also, say that any ancestor of an ancestor is an ancestor. If h is a 1-1 function from A onto B such that for all x in A, either h(x ) = f (x ) or h(x ) = g-1(x ), show that:
(a) If x has a finite, even number of ancestors, say 2n, then h(x ) = f (x ). Hints: Use induction on n.
(b) If x has a finite, odd number of ancestors, then h(x ) = g-1(x ).
(c) If x has infinitely many ancestors, then how is h(x ) chosen according to the proof of the equivalence theorem (1.4.1)?
Describe all the maximal chains
: 1. In the partially ordered set N × N with the ordering ( j, k) ≤ (m, n) iff j ≤ m and k ≤ n, consider the sequence (n, n), for n = 0, 1, 2,... . Describe all the maximal chains that include the given sequence. Do any of these chains have upper..
|
Rock climbing gear for yosemite national park
: What is the best rock climbing gear for Yosemite national park?
|
Well-ordering principle implies
: Prove, without applying Theorem 1.5.1, that the well-ordering principle implies AC. Caution: Is it clear that the possibly very large family of sets A(x ) can all be well-ordered simultaneously, so that there is a function f on I such that for eac..
|
General without the axiom of choice
: (which cannot be proved in general without the axiom of choice, to be treated in §1.5). Hint: Let B be a countably infinite subset of X \Y . Then B and B ∪ Y have the same cardinality.
|
Determining the proof of the equivalence theorem
: If x has infinitely many ancestors, then how is h(x ) chosen according to the proof of the equivalence theorem (1.4.1)?
|
How long is alexa kayak
: Jude's kayak is 10 feet and 2 inches long. Alexa's kayak is 29 inches longer. How long is Alexa's kayak
|
What is an action potential
: What is an action potential? Where does it occur and how is it propogated?
|
Subset of it has a minimal element
: Given a partially ordered set (X, ≤), a subset A ⊂ X , and an element x ∈ A, x is called a minimal element of A iff there is no other y ∈ A with y x . Then (X, ≤) will be called min-ordered iff every non-empty subset of it has a minimal e..
|
Compare and contrast the active
: Compare and contrast the active and passive immune system in the context of HIV viral infection.
|