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Use mathematical induction to show that 11 people can

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 53E Chapter 5.1

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 53E

Use mathematical induction to show that 11 people can divide a cake (where each person gets one or more separate pieces of the cake) so that the cake is divided fairly, that is, in the sense that each person thinks he or she got at least (l/n)th of the cake. [Hint: For the inductive step, take a fair division of the cake among the first k people, have each person divide their share into what this person thinks are k + 1 equal portions, and then have the (k + l)st person select a portion from each of the k people. When showing this produces a fair division for k + 1 people, suppose that person k + 1 thinks that person/got Pi of the cake where .]

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Class #7 Duress or Coercion ● Duress or coercion defense: a person can be force to be involved in a crime ○ threats must put defendant in fear of imminent death ○ threats must be such that a reasonable person in the defendant's circumstances ● Duress or coercion in murder ○ common law made no provision for the duress defense for taking the life of an innocent person ○ today, state statutes (like the common law) generally exempt the duress defense in murder cases ○ distinction often drawn between duress forcing intentional act of killing third person, and duress forcing involvement in felony in which their person is killed ● Necessity (choice of Evils) ● Necessity and Homicide ○ not typically a defense in homicide cases ○ the threat of death to A does not justify A’s killing of B ■ A is not justified by necessity to push B out of a lifeboat that will only hold 1 person ■ BUT if A has already established a place on the boat, necessity may justify blocking B from getting on the boat ○ Defense may be available where taking on life saves many ● Alibi: somewhere else in latin ○ Alibi as a criminal defense: in using the defense of alibi, the defendant is asserting that he or she physically could not have committed the crime, because at the time the crime was committed he or she was at another place ■ because jury cannot convict if a reasonable doubt exists, alibis are effective defenses ■ the burden is not on the defendant to show that he or she was not at the scene of the crime but the state can show beyond reasonable doubt that the defendant WAS at the scene ■ because it can be fabricated it must be investigated ● Double Jeopardy ○ means that a person who has been acquitted or convicted by a judge or a jury may not be tried again from the same crime ○ miss trial = not guilty or not guilty (NO verdict) ○ A different or separate offense ○ Double jeopardy: not violated in state vs. federal government bring the charge ○ retrial after a “hung” jury is not double jeopardy in a single­count prosecution ○ When it applies: ■ an earlier prosecution must progress to the point of jeopardy attachment ● in a jury trial, jeopardy attaches as soon as a jury has been empanelled and sworn ● What is the “Same” offense... Two tests ○ “Same elements test” (Blockbuger) ○ “Same Transaction Test” ● Speedy Trial ○ 6th Amendment guarantee ■ applies to both federal and state trials ○ United States v. Marion ■ right attaches only after the prosecutorial phase commences ○ Considerations ■ The length of the delay ■ reasons for the delay ■ whether the defendant made a timely assertion of his right (“or sat idly by”) ■ Whether the defendant was prejudiced by the delay ● Dual Sovereignty ● Entrapment: defense is based on official action that induces a person not already predisposed to criminal behavior ○ predisposed means “ready and willing” to commit the crime when an opportunity arises ○ deter objectionable government conduct ○ not accepted in homicide cases ○ defense requires “inducements” by officials that go beyond usual investigation ■ “repeated or persistent solicitation” ■ “coercive encouragements” ■ “aggressive persuasion” ● Statutes of limitations ○ places a limit on how long after a crime has been committed a persecution may be commenced ○ purpose is both eliminate anxiety about possibility of prosecution, and help ensure that evidence is fresh and not obscured by the passage of time ○ Not of constitutional dimension, so states and federal government are free to set limitation periods as deemed appropriate ■ many crimes such as murder or rape do not apply Chapter 9: Free Speech, Public Order Crimes, and the Bill of Rights ● the freedom to believe is absolute and the freedom to act is not ● The 1st Amendment ○ protects freedom of religion, speech, assembly, the press and the petitioning the government for redress of grievances ○ While the constitution protects the “freedom of the speech,” that protection is NOT absolute ○ must have a state actor ● Free Speech, the Bill of Rights, and Criminal Law ○ some kind of speech are not protected by the 1st amendment, and may be the subject of criminal laws ○ Laws directed at the content of speech are subjected to strict scrutiny, meaning the government must have a compelling interest in achieving some goals, and that the limit on the speech is necessary ● Criminal Laws Aimed at Content of Speech ○ Clear and Present Danger Test: speech that creates an immediate threat of harm to another may be suppressed by criminal law ○ Fighting words: words likely to incite “an immediate breach of the peace” and may be suppressed ■ higher burden for police ■ must be done face to face not over the phone or email or text ect. ■ fighting words vs. true threat (officer jason I’m going to kill you and your family) pg. 234 memorize the box ● Obscenity ○ when identified by a state, can be punished as a crime ○ defining it has been the subject of many court ruling and debates ○ In general freedom to use obscenity is not protected by the 1 amendment when: ■ the communication is described in an offensive manner depiction of explicit secual acts ■ protrals secual conduct in a patently offensive ■ the communication has no redeeming social value ○ Pronography, when identified by a state, can be punished as a crime ○ Define pornography, however, has been the subject of debates ● Defamation ○ falsity ■ defenses ○ Communicated ○ to a third party ■ not the victim (the subject of the defamatin) ○ Damages ● Inciting Unlawful Conduct ○ speech intended to produce unlawful actions is not protects ■ if purpose of speech is to incite crime, it has no claim to 1st amendment protections ■ judged by clear and present danger test ○ Must be more than mere discussion of violence or unlawful actions. Unlawful conduct must be “imminent” ● “True Threats” ○ speech intended to put another of bodily harm is NOT protected under the 1st ○ A TRUE THREAT is a serious expression of an intent to inflict bodily harm (normally seen in the charge of “communicating threats” ○ Examples ■ threats against public officials ■ threats by schoolchildren against teachers or students ■ terrorist threats ■ violation of restraining orders ● Public Nuisance Laws ○ states have PUblic Nuisance Laws which are intended to insure public safety and public order. Some regulations include: ■ Laws on loud noise ■ nuisance speech ■ riots ( 3 or more people gathered in a public place for a common purpose to disturb the peace) ● Symbolic speech non verbal expression that convey a belief or idea ○ Actions that have a purpose other than beyond communication of an idea or belief are not protected by the 1st amendment ○ symbolic speech that is disruptive can reasonably be regulated ○ in past years, states had statutes that made the burning or defilement of the American flag a crime ○ Cross burning can be a symbolic act ■ if the purpose to intimidate it can be criminalized ● “Street Crimes” and Individual Rights ○ Criminal laws have been created which regulate safey and minimize street crimes. Some city street laws include: ■ anti­cruising laws ■ truancy arrests or detentions ■ juvenile curfew laws ■ laws directed at homeless persons ■ stalking ○ in considering the balance between individual rights and public safety, the scales are tilted towards public ● The Second Amendment ○ the government is allowed to put reasonable restrictions on one's guns outside of the home ○ government may not outlaw completely ownership or possession ○ still considered reasonable that juveniles and felons can’t own firearms ● Freedom of Religion ○ the 1st amendment protects the right to practice a religion without prohibition of the government ○ freedom to BELIEVE is absolute; the freedom to act is not ■ handling poisonous snakes in public ■ child labor ■ human and animal sacrifices ○ some conduct criminal in secular settings has been permitted in religious settings ● Right of Privacy ○ in U.S constitution has ne explicit right of privacy, but it is implied in the concept of “personal liberty” found in the 14th amendment ○ my no pass criminal law that impinge upon the right, unless the laws have a reasonable, legitimate, governmental purpose and are tailored to advance that interest with the least restriction on the right to privacy ○ conducts permitted in privacy may be regulated in done in public (Abortion) ­if it is the 1st trimester before viability = woman choice ­after 1st trimester = reasonable regulations ­after viability = government can be completely outlawed but there can be an exception for the life of the mother

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Chapter 5.1, Problem 53E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Use mathematical induction to show that 11 people can