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Adapt the proof in the text that there are infinitely many

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

Solution for problem 54E Chapter 4.3

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 54E

Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 3k + 2, where k is a nonnegative inte­ger. [Hint: Suppose that there are only finitely many such primes q1,q2,···, qn, and consider the number 3q1q2···qn – 1.]

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1 Reproductive Systems Male vs Female  Primary reproductive structure in males are the testicles. o Occurs in pairs.  Internally divided into sections. (if dissected, would look internally same as an orange)  Each section (lobe) has a seminiferous tubules.  Each tubule is roughly 3 feet long.  Seminiferous Tubules are where sperm production occurs. Epididymis o Extension of the tubules on the outside of the testicle.  This is where sperm is stored for brief periods of time  Vas deferens  Connects to the epididymis moves sperm from the testicle to the urethra. o (Urethra) This is the only structure shared by the reproductive and urinary system in males. o No structures are shared between the female reproductive and urinary systems.  Production of Semen  Seminal vesicle­Near the prostate gland toward the end of the Vas Deferens.  Prostate gland: production of semen. o Surrounds the urethra, located at the base of the bladder o The prostate continues to grow throughout life.  Has a very fibrous outer covering which restricts oenlargement o As it grows, it may restrict flow within the urethra making it difficult to completely empty the bladder. o Prostate is prone to develop cancer, easily treatable if caught early.  Cowper’s gland 2 o Located at the base of the prostate o Connects to the urethra Urethra:  Drains the bladder and facilitates the introduction of sperm into the female reproductive tract. Penile Shaft  Made up of 2 types of tissue (Corpus Cavernosum and Corpus Spongiosum)  Collectively known as erectile tissue o When stimulated, these tissue fill with blood causing an erection  Interstitial Cells  Found in the testicles  Major function: production of testosterone  Sperm:  main regions o Tail: used for swimming o Midpiece: has numerous mitochondria  Mitochondria provide ATP needed by tail o Head region:  Nucleus (needed for fertilization)  Acrosome: package of digestive enzymes  Enzymes are used to degrade the outer portion of an ova (egg)  Seminal Fluid (Semen)  structures that produce semen o Testicles: contribute sperm to the semen o Seminal vesicle & Prostate & Cowper’s Gland: Produce the rest of the semen  Contained in it: Sperm, Alkali Solution (Used to neutralize the pH of the female reproductive system.  Sperm are most active in a neutral environment (pH around 7)  Female reproductive tract tends to be acidic o Fructose: produced by one of the 3 glands. o Broken down by the sperm in order to produce ATP 3 o Prostoglandins:  Hormones that cause uterine contractions  Force sperm higher into the female reproductive tract  External Genitals Testicles: o Surrounded by scrotal sac  Scrotal sac expands and contracts under the influence of temperature  Suspended outside of the body (usually mammals)  Sperm production is greatly influenced by temperature  98.6 is too hot for functional sperm  93 is optimal temperature  Female reproductive system ovary: Primary reproductive structure oProduces Ova oSolid Structure oProduces some hormones (estrogen) o The ovaries take turns during each uterine cycle Oviduct:is a tube that is loosely attached to the ovary o Captures an ova when released conducts it towards the uterus is where fertilization occurs Fallopian tubes; uterine tubes o At the tip of the oviduct where it loosely attaches to the ovary is fingerlike growths called fimbriae o Are in constant motion and create a “water” current which forces any released ova into the oviduct  Uterus oPear shaped organ, mostly smooth muscle, sits on top of the urinary bladder, where development will take place if fertilization occurs Vagina (Vaginal Canal) oFacilitates intercourse oreceives male penis oserves as the birth canal olined with erectile tissue Cervix oJunction between the vagina and the uterus 4 External Genitals: oClitoris  Female organ of arousal; made up of erectile tissue, almost identical in structure to the male penis. oLabia Majoria and Minora:  Folds of skin that cover the vaginal opening  produce lubricants during intercourse  Menstrual Cycle (Uterine Cycle) Average length 28 days (can run anywhere from 18­40 days) oDays 1­5:  Menstruation, inner lining of uterus is discharged o Days 6­13  Proliferation stage (build up) o Day 14  Ovulation (ova released from ovaries) o Days 15­28  Secretory phase, uterus prepares for implantation o Birth Control: o Abstinence 100% o Vasectomy almost 100% o Tubal ligation (uterine tubes clipped) almost 100% o Oral contraception (pill) almost 100%  When tracked, it’s actually closer to 85% because the pill needs to be taken approx same time every day o Contraceptive implants 95% o Contraceptive injections 99% o Intruterine Device (IUD) 90%  Device inserted into the cervix; usually made of copper or some kind of metal.  does not prevent fertilization, just implantation o Diaphragm 90%  A device that covers the cervix; prevents sperm from reaching the ova o Cervical Cap 85%  Plugs the cervix; prevents sperm from reaching the ova o Male Condom 85%  sleeve that fits the penis and traps sperm; the most effective method at preventing STD’s o Female Condom 85% 5  sleeve that fits into the vagina and traps sperm very good at preventing STD’s o Coitus interruptus 75%  removal of penis before ejaculation; couples that are attempting to get pregnant have only a roughly 25%  success rate and 25% failure rate o Jellies, creams and foams 75%  chemicals that have spermicidal effects (kills sperm)  are intended to be used with other forms of birth control o Natural family planning 70%  plan days of ovulation, avoid sex several days around that time o Douche 70%  Contains spermicidal chemicals; while flushing some sperm out and some higher o Sexually Transmitted Diseases  AIDS caused by a virus; treatable but at present not curable  acquired from an exchange of body fluids the immune system degenerates (targets helper t­cells)  victims tend to die from rare type of diseases  Genital Herpes  Caused by a virus; treatable but not curable  open sores on the genitals  tend to be self­healing  tend to reoccur  closely related to the virus that causes cold sores  active outbreaks (individuals) tend to be caused by stress  Genital Warts  Caused by a virus, warts develop on the genitals  treatable but not curable;  they tend to reoccur  in females, there appears to be a link between genital warts and certain types of uterine/ovarian cancer Gonorrhea  Caused by bacteria  easily curable with antibiotics 6  one of the more common types of STD’s  Symptoms: o white discharge from the penis/vagina o painful urination o in men: scar tissue can develop in the urethra may lead to a complete blockage medical intervention is required o in females: symptoms seem to go unnoticed  scar tissue develops in the uterine tubes  leading to sterility o at one point, was the most common cause of sterility medical intervention is required Chlamydia  Probably the most common type of STD but least reported  most people are a­symptomatic  some people develop flu like symptoms but recover quickly  some develop painful urination but recover quickly  Easily treated with antibiotics  Has a very weak immune response  Predisposes a person to contracting other STD’s o Most people diagnosed with an STD, have chlamydia (85%)  Syphilis  caused by bacteria  easily treated in the early stages (antibiotics)  impossible to treat in later stages  Stage 1: an (painless) ulcer develops on the genitals  usually heal within a couple of weeks and may leave a scar  Stage 2: a rash develops on the palms of the hand/soles of the feet  usually occompanied by flu like symptoms; may occur several months after the ulcer heals; rash clears in a couple of weeks  Stage 3: development of gumas (large weeping ulcers that develop throughout the body)  The worst ones are the ones that develop on the internal organs (lead to death)  untreatable 7  Parasitic infection  Initial symptoms can last up to 9 months  Secondary can last up to 30­40 years  o Fertilization: union of a sperm and an ova which forms a Zygote.  Only 1 sperm penetrates the egg and the nucleus of the sperm and ova combine o Takes place in the Oviduct (fallopian tubes)  Requires hundreds of sperm/but only one penetrates the ova  Sperm: o Head: acrosome, nucelus o Middle piece: mitochondria for power o Tail: swimming  Ova: o Corona radiata: thick outer layer  Consists of cells from the ovary o Zona pellucida: thickened middle layer o Plasma membrane  Is not degraded during a fertilization event o The events of fertilization: o 1: thousands of sperm reach the ova and release the contents of their acrosome o 2: enzymes from the acrosome begin to degrade the corona radiata and zona pellucida o 3: eventually, one sperm comes in contact with the plasma membrane the remaining events take place within microseconds.  a: plasma membrane separates from the zone pellucida this creates a gap making it difficult for additional sperm to enter  b. a gap created fills with fluid  c. the interior of the ova begins to spin o Sperm and ova are unique because they each have a haploid nucleus so that when they combine they maintain the original chromosome count  Haploid means half. o Development after Fertilization: o Cleavage: the zygote begins to divide without increasing in size, the cells numbers double after each division 8 (the cell size decreases by half during each division) o Morula: solid ball of cells produced by cleavage events (mulberry) o Blastula: cells from the center of the morula migrate to the outside leaving a hollow ball  some cells remain in the cavity o Gastrula: cells invade the space between the endoderm and the ectoderm (forming endoderm and ectoderm)  Cells reinvade the hollow space left by the blastula.  This forms 2 layers of cells  Inner layer: endoderm o will eventually produce most of the internal organs  Outer layer: ectoderm o will eventually form the skin and nervous system  Toward the end of gastrulation, the cells invade the space between the endoderm and ectoderm (these cells form mesoderm)  Mesoderm will eventually produce skeletal and muscle tissue o Neurula: notochord develops which will eventually be replaced by the spinal cord o Germ Layers:  Ectoderm: skin, brain and neurons, linings of the nose mouth and anus  Mesoderm: muscles, connective tissue, reproductive organs  Endoderm: digestive tract, glands, bladder (other internal organs) o Implantation  Zygote embeds into the uterine lining oPlacenta:  A huge capillary bed (exchanges w/tissues)  Forms from fetal cells/blood vessels from the fetus intertwine with mothers blood vessels (forming a huge capillary bed) to provide nutrients and remove waste from developing embryo.  Is only present during pregnancy  Delivered after the baby (afterbirth)  The placenta can develop anywhere within the uterus.  Umbilical cord attaches from the fetus to the placenta o Embryonic Development o 1st Stages: First and second months of pregnancy 9 o Week 1: pre­embryonic development implantation has not occurred, cleavage events occurring, nutrients are supplied by yolk, and blastula formation occur  Cell mass may split to form identical twins  Week 2: Cell mass arrives at the uterus  implantation occurs  placenta begins to form  Tissues are being produced  Week 3: Nervous system begins to develop (first system) o Circulatory system begins to develop  Week 4: the appearance of a tail is present on embryo o Head is much larger than rest of embryo o Limb buds are present o Eyes, ears, and nose all begin to appear o Heart beats o Liver produces blood cells oSecond Month: oArms and legs become more developed oStart getting fingers and toes  By end of second month all major organ systems have developed o Fetal Development o Third and Fourth month:  Head growth slows (does not stop)  Eye­lashes and eye­brows, hair on head, fingernails and nipples begin to appear  Bone begins to replace cartilage  Heartbeat may be heard  Approximately 6 in/6oz  Amniotic Sac: filled with fluid, prevents drastic temp change, protects against mechanical damage. o Fifth through Seventh month:  Movement may be felt  Lanugo covers the baby (fine downy hair covering the entire body)  Eyelids open 12 in/3 pounds  Also covered with Vernix Casiosis (waxy substance looks like cheese) 10  Prevents the fetus from becoming water­logged o Eighth and Ninth Month  Head begins to point down towards the cervix  Growth of Fetus  21 inches/7.5 pounds o Birth o Stage 1: Mucus plug, which has been at the cervix, is expelled  Amniotic sac breaks (water breaking)  Cervix dilates to 4 inches o Stage 2: Contractions every 1­2 minutes  The uterus began contracting somewhere in the second/third month  (called braxton hicks contractions)  Crowning: Baby’s head appears in birth canal  Baby is expelled head first  Umbilical cord is cut after baby begins to breath normally o Stage 3:  Delivery of the placenta

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Chapter 4.3, Problem 54E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. The full step-by-step solution to problem: 54E from chapter: 4.3 was answered by , our top Math solution expert on 06/21/17, 07:45AM. Since the solution to 54E from 4.3 chapter was answered, more than 325 students have viewed the full step-by-step answer. This full solution covers the following key subjects: Primes, infinitely, nonnegative, form, ger. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. The answer to “Adapt the proof in the text that there are infinitely many primes to prove that there are infinitely many primes of the form 3k + 2, where k is a nonnegative inte­ger. [Hint: Suppose that there are only finitely many such primes q1,q2,···, qn, and consider the number 3q1q2···qn – 1.]” is broken down into a number of easy to follow steps, and 51 words. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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Adapt the proof in the text that there are infinitely many