Algebra

 

1: Show that: There is a family of cardinality bigger than continuum of subsets of R
that is ordered with respect to inclusion.
2: Verify: Any countable ordered set is similar to a subset of Q ∩ (0, 1).
3: Any countable densely ordered set without smallest and largest elements
is similar to Q.
4: Any countable densely ordered set is similar to one of the sets Q∩ (0, 1),
Q∩[0, 1), Q∩(0, 1], Q∩[0, 1] (depending if it has a first or last element).
5: There is an uncountable ordered set such that all of its proper initial
segments are similar to Q or to Q ∩ (0, 1].
6: Verify: There is an uncountable ordered set which is similar to each of its uncountable
subsets.
7: What is the cardinal a0 • a1 • • • if the ai’s are positive integers?
8: Prove: (Fundamental theorem of cardinal arithmetic) For every infinite cardinal
κ we have κ2 = κ.
9: Show that: If at least one of κ > 0 and λ > 0 is infinite, then
κ + λ = κλ = max{κ, λ}.
10: Show that: The supremum of any set of cardinals (considered as a set of ordinals) is
again a cardinal.

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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