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Homework 7 (419):
1: Let X be a set of infinite cardinality κ, and call a set Y ⊂ X “small” if
there is a decomposition of X into subsets of cardinality κ each of which
intersects Y in at most one point. Then X is the union of two of its “small”
subsets.
2: Show that: If X is of cardinality κ ≥ ℵ0, then the following sets are of cardinality κ:
a) set of finite sequences of elements of X,
b) set of those functions that map a finite subset of X into X.
3: Show that: An infinite cardinal is regular if and only if κ is not the sum of fewer than
κ cardinals each of which is less than κ
4: Show that: A successor cardinal is regular.
5: Which are the smallest three singular (i.e., not regular) infinite cardinals?
6: Show that: If α is the cofinality of an ordered set, then α is a regular cardinal.
7: Show that: (Each part is one point)

 

(a) Every element of an N-set is an N-set.
(b) If x is an N-set, then y = x ∪ {x} is an N-set, and if z is an N-set
containing x, then y ⊂ z.
(c) If x is an N-set, y ∈ x, then y is an initial segment of x.
(d) If x is an N-set and Y ⊂ x is one of its initial segments, then Y is an
N-set, and either Y = x or Y ∈ x.
(e) If x, y are N-sets, then x = y or x ∈ y or y ∈ x.
(f) For N-sets x, y define x < y if x ∈ y. Then this is irreflexive, transitive
and trichotomous. Furthermore, if B is a nonempty set of N-sets, then
there is a smallest element of B with respect to < (“well order”).
(g) If x, y are different N-sets, then they are not similar.
(h) Every well-ordered set is similar to a unique N-set.
8: Show that: There is no infinite decreasing sequence of ordinals.
9: Show that: Arbitrary infinite sequence of ordinals includes an infinite nondecreasing
subsequence.
10: Show that: Addition among ordinals is monotonic in both arguments, and strictly
monotonic in the second argument. The same is true of multiplication
provided the first factor is nonzero.

 

 

 

Sample Solution

duction that have been conducted in the natural sciences that are mainly the product of an individual. Albert Einstein was a German theoretical physicist who majorly contributed to modern day physics through his theory of relativity and other findings. In 1905, Einstein was working at the patent office doing creative work. He began to produce at least four revolutionary articles in 1905 by himself. The first essay Einstein published was an application of the quantum theory, a conjecture founded by physicist Max Planck. This theory was related to light to explain the photoelectric effect to expand on the ideas of Planck. The second article was about atoms and electrons analyzing motion based on the findings of Robert Brown. The other two articles were based on two principal theories of physics by Isaac Newton. Even though the articles were written and analyzed by Einstein alone, each paper was based on the findings of another demonstrating collaboration. This shows that the production of knowledge can be individual but must be collaborative to some extent. Similarly, there are studies of knowledge production being an individual task in the human sciences. Psychologist Roger Sperry performed multiple experiments animals and humans in order to find the function of each hemisphere of the brain in regards to behavior. In the 1950s and 1960s, he studies and severed the corpus callosum to discover split brain, the idea that the hemispheres of the brain function independently. In the late 1900s, he performed more split brain research at the California Institute of Technology. Most of his research revolved around his dissections of the corpus callosum. Since he was conducting these studies by himself, many years later he found that the left hemisphere relates to understanding language while the right hemisphere relates to recognition and not articulation. Because his research was revolutionary and the product of an individual, Sperry received the Nobel Prize for his work. In order to further expand on his discoveries, other psychologists have expanded on Sperry’s work to evaluate the right and left hemispheres as seen in the Embryo Project at Arizona State University. Through shared knowledge, this discovery along with findings were solidified. Since other psychologists have utilized the split brain research, there has been some collabor

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