QUESTION 1

QUESTION1

Let forlet be an increasing function (i.e. if).Prove that if the sequence converges, to then isincreasing, and that if is continuous then the convergence is uniform.

  1. We first seek to prove that if the sequence converges to then is increasing.

Let such that. Since,thensuchthat ,

Whenever

And

Whenever

If we let then we have and.Implying that.Since we can make as small as we want, holds.

  1. Secondly, we have to show that if is continuous then the convergence is uniform.

We will prove by contradiction. That is, we assume that iscontinuous but the convergence is not uniform. This implies that such that where isa function of and where for.Due to being continuous, diverges to as.The fact that leads to a contradiction if we consider the value of suchthat and.Therefore isuniformly continuous.

QUESTION 2

The graph of is the set of

  1. If is connected and is continuous, prove that the graph of is connected.

We’ll have to show that the image ()of a connected set ()by a continuous function is connected.

We’ll proceed to prove the contrapositive. That is if ()is not connected then ()is not connected. If ()is not connected, then with, , (i.e. they are disjoint) and .

Considerand note that these two separateby definition of open sets. We know that since and similarly.Again since we have that.

Finally, sinceimplies implying that or.Thus is not connected.

  1. Give an example to show that the converse is false.

Let.Now but at.Therefore such that

  1. If is path-connected and is continuous show that the graph is path-connected.

Consider Then, since isconnected a continuous curve such that and.Then the curve is continuous and maps.

  1. What about the converse?

The converse is not necessarily true if we consider to be a discrete space with more than one point (for it to bedisconnected, it has to have more than one point) that maps to theconnected set.

QUESTION 3

  1. Prove that a continuous function, all of whose values are integers, is constant provided that is connected.

We will prove the contrapositive. Suppose such that (i.e. are distinct values of).Then we have andare both non empty disjoint open sets since all values come from adiscrete space of integers such that since andare arbitrary integers. This implies that if is not constant, then is not connected (the contrapositive).

  1. What if all the values are irrational?

Similarly, will have to be constant since is only connected if and only if is single valued. This is due to existence of the rational numbersthat allow to be written as a union of disjoint open sets when.

Question #1

Question#1

Numerouselements of culture are common to various societies. The firstelement is symbols. A symbol can be anything that is recognized bypeople of a culture and have a particular meaning. For example, mostthe cultures have flags that are symbols and have a particularmeaning. Humans make symbols every day, and sometimes they manipulatethe existing symbols to change their meaning. Another element that iscommon to many societies is language. Every culture has a uniquelanguage. Language is a cultural transmission of symbols and allowspeople to communicate. Language can be written, and different groups,write from left to right and others they write from right to left.Our bodies contain different symbols and language is one of them.Every culture has their values and beliefs these two are importantelements. People in a particular culture share values and beliefs.Values are a combination of specific, thoughts, beliefs and ideasthat individuals hold to be true. For example, many countries believein providing equal opportunities for both female and male citizens.People value self-expression and individualism and believe thatcountries should ensure citizens have higher incomes to support theirneeds.

Normsare an important part of a culture. Norms according to the author arebehaviors that are accepted by the societies and have great moralsignificances. According to Graham Summer, there are many norms inculture, but some are vital than others. People are always concernedabout what the norms when they are in their regular activities.Graham characterization of norms includes mores and folkways. Thenorms that are vital to a society and are widely observed arereferred to mores. Graham continues to explain that norms that peopleare not concerned with or pay less attention are known as folkways.For example, one of the norms that can be characterized as a folkwayis the concerns of proper dressing. A norm that is characterized as amore, for example, can be the taboos many cultures have about sex.These taboos specify the age when individuals should start sexualactivities.

Anotherelement of culture is social controls. People should understand thatMores and Folkways are the basic rules followed every day, but socialcontrol regulates both of these elements. It regulates people’sbehavior and thoughts. Social control as an element can simply bedefined as the regulation of people’s norms. There are manycultural diversities for example, high and popular culture. Thecultural patterns bring these diversities and according to thesociologist, high culture can be defined as cultural wilderness andimpersonators that display subcultural patterns. Subculture is smallgroups of culture who have different subcultural patterns. Popularculture is cultural pattern that is accepted by numerous cultures.Multiple culturalisms are pattern why society has many culturalpatterns from numerous different societies or groups. For example, inAmerica, there are many people of different cultures from differentcountries. Multiculturalism increases diversities, and according tosupporters, it strengthens academic achievements of children.Ethnocentrism is a multicultural diversity created by individuals ofdifferent ethnic groups. According to the sociologist, Afrocentrismemphasizes and promotes African cultural patterns in other cultures.Cultural changes are seen in cultural patterns and systems. Forexample, in college during the 19 century, students studied andthought of creating families after but nowadays students are moreconcerned about success in life and making money. A change in onedimension of a cultural pattern and dimension can cause changes inothers. For example, women in the previous year’s always thoughtabout being mothers, but nowadays they think of earning money andraising their social status.

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Question#2

Hunting,gathering societies

Thisis the simplest society compared to any other. The use simple toolsand gathers vegetation for food. These societies started from thebeginning of human species` appearance. It started 3 million yearsago and was found all around the world. Nowadays, only a fewsocieties are involved in hunting and gathering techniques. Thesesocieties are the Bushmen of southwestern America, aka, aboriginal ofAustralia and the pygmies of central Africa. These societies spend alot of their time hunting and gathering because there is littleability to control the environment. The features that these twosocieties have evolved are the use of the modern tool for farming andhunting. Joining other societies is also a feature of evolution. Thesocial structure that would have possibly existed in these twosocieties is a role. For example, in this society, every person had arole in his or her. Men used to hunt, and women used to cook and helpwith farming.

Horticulturaland Pastoral Societies

Thissociety started approximately 12,000 years ago. Their numbersdecreased after about 3000 B.C.E due to evolution. These societiesinvolve two groups. The first is the horticultural society that usedhand tools for cultivation. The pastoral societies are the secondgroup and involved in the domestication of animals. They hadsettlements of hundreds of people, and they connected to thousands ofpeople through trading activities. The pastoralists are nomadic, andthe horticultural society is involved in making small permanentsettlements. This society has evolved the pastoralist have considereddomesticating the animal. The horticultural society has consideredusing new developed hand tools and machines. The social structureinvolved in this type of society is a role. Every individual in thesocial had a particular role. The men’s role was to protect thefamily and women to nurture them.

Agrariansociety

Thissociety began approximately 5000 years ago and decreased in hugenumbers in the recent years. They used animals to plow their farms.This group included millions of people. In the cities, there aresmall groups of the agrarian society. One example of these groups isthe Egyptians during the period of the great pyramid construction.There are numerous agrarian societies all around the world. Thissociety has evolved and considered to use new sources of energy topower the agricultural machines. They do not use an animal to plowthe farms, but use machines since they are faster and easier to use.The societal structure of agrarian is possibly social groups. Itinvolves family, friends, and couples as social groups.

IndustrialSocieties

Thissociety began in approximately 1750 up to the recent years. They usedadvanced sources of energy and mechanized production. They involvemillions of people, and the population is concentrated in the cities.This society is involved distinct groups of political, religious,family and economic systems. Most of the people in this society areconcentrated in Japan, Europe, Australia and North America. Thissociety had the power to change and alter the environment. Thechanges became visible even before the 18thcentury. The features that show this group has evolved are the userecent sources of energy to power their machines. The socialstructure of this society would possibly be status. There are allkinds of status in this society. For example, the types of statusesare rich and poor, disabled and educated versus the uneducated.

PostindustrialSocieties

Thissociety has emerged in the recent decades due to the inventions ofthe computer. The computers support the economies information. Thereare millions of people in this society, and they are concentrated inthe cities. Most of the people in the industrial societies are nowentering the postindustrial society. The postindustrial society hasevolved in the recent years. They use the computers to generate newinformation. The social structure of this society is possiblyinstitutions. There are many different institutions in this society. For example, the colleges and universities can be used todifferentiate the society

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Question 1

TheSupreme Court’s case of Lamb against California decided on January7, 1963, found on page 234 of volume 371 in the United StatesReporter is correctly cited as Lampv. California,371 U.S. 234 (1963)Question2

Thefederal district court fortheDistrict ofMassachusettsgets bound by the following courts’ decisions:

  • U.S. District Court, District of Massachusetts

  • First Circuit Bankruptcy Appellate Panel

  • U.S. Court of Appeals for the First Circuit

  • Supreme Court of the United States

Question3

Thefollowing is an order from the most binding to the least bindingsources of law within the legal system:

  • Federal Constitution

  • Federal Legislation

  • State Constitution

  • State Legislation

Question4Whenthe decision of a court is mandatory, then it must be followed byanother court in the same jurisdiction or legal order. At times thisis not the case, a typical situation where decisions from one stateare highly persuasive in another is when they share a particulardoctrine. The sharing of a doctrine may be due to geographic,historical or demographic similarities that have led to the adoptionof similar principles. A Texas courts may find decisions of Wisconsincourts in marital property cases quite persuasive because both statesadhere to community property law (Hanrahan, 2012). A case, however,could be mandatory even though it did not originate from your region.When the decision in a federal court covers a point of state law,therefore the state’s interpretation of that law is mandatory inthat federal court. State courts at times also decide issues offederal law, but federal courts do not bind them, it is only the U.SSupreme court that is mandatory on state court when it decides on thefederal legislation. WorkcitedHanrahan,C. (2012). Legalsystem.Detroit: Greenhaven Press.