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Suppose that a and b are integers. a 11 (mod 19), and b 3

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

Solution for problem 14E Chapter 4.1

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 14E

Problem 14E

Suppose that a and b are integers. a ≡ 11 (mod 19), and b ≡ 3 (mod 19). Find the integer c with 0 ≤ c ≤ 18 such that

a) c ≡ 13a (mod 19).

b) c ≡ 8b (mod 19).

c) c ≡ a - b (mod 19).

d) c ≡ 7a + 3b (mod 19).

e) c ≡ 2a2 + 3b2 (mod 19).

f)  c ≡ a3 + 4b3 (mod 19).

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PLANTS (important vocab terms are highlighted in yellow) Graham Kulig Dr. Vandermast’s Biodiversity class ● Dominant primary producers ● Important source of oxygen and food for aquatic organisms ● Hold soil in place ● Moderate extreme temperatures ● Land plants were first multicellular organisms with most of their tissues exposed to ai(pay attention to adaptations of living on ) y land​ ● Green algae are closest living relatives to plants ● Transition from aquatic to terrestrial life occurred when land plants evolved from green algae Non­Vascular Plants: Bryophytes: ● Possible test q: what are the 3 reasons bryophytes are short ● All seedless and non­vascular plants most basal lineage of land plants and the gametophyte is the dominant phase of the life cycle. ● Grow low to ground ● No ​Lignin ● Rhizoids anchor plant to soil/rock ● Need water for reproduction, flagellated sperm swim to eggs ○ Spores are wind­dispersed Liverworts: ● Grow in dense mats on forest floors ● Have ​ cuticl (prevents loss of water), and some have pores ● Asexual reproduction occurs through ​ gemmae​ , which are knocked off parent plant by rain or wind Mosses​ : ● Very hardy, grow in extreme environments, can dry out, and then rehydrate itself ● Less than a few cm in height ● Sexual reproduction cannot occur via ​ self fertilizat (Antheridia produce only antheridia) ● Foot seta, capsule, periculum ​are allsporophyte Hornworts:​ ● Sporophyte has hornlike structure, which is why it’s called a hornwort Vascular Plants: ● Paraphyletic group ● All have vascular tissue reinforced with lignin ● Depend on water for reproduction ● Sporophyte is dominant phase Lycophyta:​ ● Fossil club mosses were huge, extant ones aren’t ● Most ancient plants with roots ● Unusual leaves called microphylls ● Asexual reproduction by gemmae or fragmentation Whisk ferns: ● No fossil record, live in tropical regions ● Lack leaves and roots ● some gain nutrients by fungi Horsetails (sphenophyta)​ : ● Flourish in waterlogged soils ● Asexual reproduction, fragmentation or sprouting from rhizome Ferns (pteridophyta): ● Most species rich seedless vascular plants ● Abundant in tropics ( few cm­20m big) ● Ferns are ​ homosporous ● Leaves are called fronds ● Sporangia found in clusters called sori on the underside of the fronds ○ Fronds with sori are called ​ sporophylls​ (spore bearing leaf, phyll means leaf) ● Monoecious: ​ (one house) Have both male and female reproductive structures in the same house ● Diecious: ​ Some individuals are male, some female, never both on one Seeded Plants Gymnosperms​ : (naked seed) do not develop in enclosed structures Angiosperms:​ (vessel seed) Seeds develop insidcarpel ● Fossil record for green algae dates back to 700­725 mya. ● Land plants: 475 mya ● Features ○ Cuticle: waxy substance that helps retain water ○ Spores are protected by ​poropollenin ○ Stomata​ (have guard cells) ○ Water conducting tissue (vasculature) ○ Leaves 5 groups of gymnosperms: 1. Cycadophyta 2. Ginkgophyta 3. Gnetophyta 4. Pinophyta 5. “Other cone bearing species” Plant facts: ● Very first endosymbiotic event: prokaryote engulfed cyanobacterium ● Land plants evolved from green algae ● Land plants are monophyletic ● Bryophytes are the earliest branching groups among land plants (most ancient) ● Non vascular plants from a grade, meaning they’re paraphyletic ● Seeded plants are monophyletic Why grow on land ● Light and CO2 are more plentiful on land Why grow upward ● Better access to light! ● 2 problems ○ Wind, gravity ○ Transporting water ● Lignin is a structural polymer. It is the defining feature of vascular cells ● Tracheids​ have: ○ A secondary cell wall with lignin and cellulose primary cell wall ● Most specialized water conducting cell: V ​essel elements Plant adaptations: (All good vocab to know!) ● Cuticle (keeps water in/out), or if not vascular, pores. ● Stomata: pores that have guard cells that open and close ● Tracheids: very first vascular cells. All plants have these, although some evolved convergently! ● Vascular tissue: individual hollow cells that help mosses get water ● Vessel elements: shorter wider cells stacked like straws that can move a lot more water. Present in angiosperms and gnetophytes Reproduction: ● Sporopollenin​ encases seeds that prevents drying out ● Embryo is nourished by mother Gametes in protected structures: Gametangia: ​ is an organ or cell in which gametes are produced Antheridium:​perm producing region on bryophytes Archegonium​: egg producing region on bryophytes Bryophytes Ferns and Gymnosperm Angiosperms Allies s Seeds No No Yes Yes Vascular tissue No Yes Yes Yes ALT of gen. G S S (G dep. on S (G dep. on S S) Vessel elements No No (just Yes gnetophytes) Water for Yes Yes No No reproduction Tracheids No Yes Yes Yes Pollen No No Yes Yes Bryophyte gametophyte structures: ● Thallus ● Rhizoids ● Antheridium ● Archegonium ● Calyptra Bryophyte Sporophyte structures: ● Foot (where transfer cells are) ● Seta (stalk) ● capsule/sporangium Ferns and allies Gametophyte: ● Thallus ● Rhizoids ● Antheridium ● Archegonium Ferns and Allies Sporophyte: ● Fronds ● Microphylls ● Sporophyll ● Sori ● Sporangium ● Strobilus Retaining and nourishing offspring ● Embryophyta:​ the retention of embryo was a key event in land plant evolution ● Transfer Cells:​ Located at foot ● Sporophyte dominated life cycle Changing Trends in Life Cycles: In seed plants, the sporophyte is dominant and the gametophyte is dependent on sporophyte (pollen) In mosses, gametophyte is dominant Alternation of Generations: Transition from gametophyte dominated to sporophyte dominated was HUGE Why Diploid cells can respond to varying environmental conditions more efficiently than haploid cells can, particularly if the individual is heterozygous Heterospory: ● Production of 2 different kinds of spore producing structures ● All bryophytes and most ferns and allies are homosporous. They produce one kind of spore ● 2 kinds of spores: M​icrosporangia and macrosporangia ○ Micro are sperm, macro are egg Monecious= ​ sexes on same structure Diecious​=sexes on different structures Pollen​: ● Evolution meant that heterosporous plants no longer needed water for reproduction Gymnosperms: Redwood group ● World’s “largest” plants ● Conifer, produces cones ● Excurrent vs decurrent tree shape ○ (Pines, Oaks. Excurrent are better adapted for snow) GYMNOSPERMS ● Heterospory:​ The production of spores of two different sizes and sexes by the sporophytes of land plants. ● When you have one plant with both micro (male) and megasporangia (female), it’s bisexual, and monecious ● angio=vessel gymno=naked ● Gametophyte dependent on sporophyte in gymnosperms ○ (sporophyte gives nutrients to seeds, which is an expensive process!) ○ DIFFERENT FROM BRYOPHYTES (non­vascular plants) (sporophyte dependent on gametophyte) FLOWERS ● All angiosperms ● Most diverse land plants (250,000+ species of plants on earth are angiosperms) ● Perfect flowers contain 2 important structures: ○ Stamen​ ­contains anther (like antlers) ○ Carpals​ ­(Middle structure that contains ovary) ● Double fertilization: A pollen grain contains 2 cells ■ Tube cell (grows into tube) ■ Generative cell­divides by mitosis and produces 2 sperm cells ● Sperm travel down tube...One sperm fuses with egg= zygote. Second sperm fuses with 2 nuclei to form a triploid (3n) nutritive tissueendosperm ● Pollination: ○ Stamens and carpels become enclosed by ​ sepals and petals ● Fruit: ​ A structure that is derived from a ripened ovary ● Angiosperm Radiation: ○ Diversification is associated with ■ Vessel elements ■ Flowers ■ Fruits Monocots and Dicots and Eudicots ● Monocots have ​ cotyledon​ (photosynthetic)​ , the first leaf, while dicots have 2 ● Monocots have scattered vascular tissue, in Dicots they’re arranged in a circular pattern ● Monocots have petals in multiples of 3, dicots have 4 or 5 Key lineages of Green Algae and Land Plants: ● Synapomorphy: Primary symbiosis ● Important primary producers ● Lichens are a mutualism between fungi and green algae ● All angiosperms have tracheids and vessel elements ● Supports all food chain systems ● Look at slide of ​ all derived traits Plant Tissues and such: Root system vs. Shoot system ● Each plant organ has dermal, vascular, and ground tissues​ to form aTissue system ● Cuticle:​ a waxy coating that prevents water loss. This is why it doesn’t help to water a plant's leaves ● Periderm ​is the cork layer of tree ● Trichomes​ :are outgrowths of the shoot epidermis and can help with insect defence Vascular system: Xylem​=wood=dead straws! Conveys water and dissolved minerals upward via co/adhesion Phloem​ transports organic nutrients (from photosynthesis) Specialization of cells: ● Parenchyma ○ Thin flexible walls ○ Least specialized ○ Can divide and differentiate (kinda like fat cells) ● Sclerenchyma ○ Rigid because of lignin ○ When functioning, they are dead ○ 2 types: ■ Sclereids ■ Fibers ● Tracheids ○ Water conducting cells, moves through pits ○ Found in xylem ● Vessel Elements ○ Straws align end to end ○ Way bigger than tracheids (Tracheids=Tunnel) ● Phloem ○ All alive at maturation ○ Only composed of a large central vacuole essentially, but companion cell keeps phloem cells alive Plants have a hierarchy of organs, tissues, and cells ● Organs: Roots, stems, leaves ● Organized into root system and shoot system Roots: ● Anchor the Plant ● Absorb Minerals and water ● Storing organic nutrients ● Taproot​ : one main vertical root that gives rise to lateral roots (branch roots) ● Absorption begins at the ​ root hairs Stems: ● Stem is an organ consisting of ○ Alternating nodes (buds) ○ Internodes=distance between nodes ● Axillary bud=is a structure that has the potential to form a lateral shoot or branch ● Apical​ bud=terminal bud, causes tree to grow. Is at the top! ● Modified stems ○ Rhizomes Leaves ● Leaf is the main photosynthetic organ of most vascular plants ● Consist of a blade and a stalk called the petiole which joins the leaf to a node of the stem Meristems generate cells for new organs ● Meristems​ are perpetually embryonic tissue and allow for indeterminate growth, like stem cells ● Apical​ meristems​ are located at the tips of roots and shoots and at the axillary buds of shoots. Used for elongation (primary growth) thickening is secondary growth ● Vascular cambium ○ Adds layers of vascular tissue called xylem (wood) and secondary phloem. Located near the outside of the tree, closer to xylem ● Cork cambium ○ Replaces the epidermis (bark) to make the plant thicker. Located on very outside of tree. ● Initials ○ New meristem cells ● Derivatives ○ Becomes specialized cells IMPORTANT THINGS TO REVIEW FOR TEST: ● Double fertilization ● Adapting to live on land ● Gemmae cups (asexual reproduction) ● Monecious vs diecious (sporophyte) ○ Refers to spore of a plant ○ Monecious=male and female reproductive structures on same plant (antheridium +archegonium on plant) Heterosporous vs homosporous (has to do with gametophyte) ● Heterosporous=2 different spore types, one grows into male gametophyte, the other grows into female ● Homosporous=One spore type that produces a bisexual gametophyte Rhizobial bacteriaL: Fix N Mycorrhizal: fungi and plant mutualism Heartwood, sapwood, vascular cambium (one cell thick, make xylem and phloem), Monocot/dicot Xylem: Transport water. Hollow cells. Passive, no energy required. Phloem: Must use atp to move nutrients (companion cells, sieve tube elements) How to tell difference​ COLOR.​ Xylem have big open cells, dyes purple/pink because of lignin. Phloem is gray. Diff between sclerenchyma and parenchyma Liverwort sporophyte pic Identify flower parts No leaf arrangement or “twig stuff” either Bonus q: GINGKO

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Chapter 4.1, Problem 14E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

The full step-by-step solution to problem: 14E from chapter: 4.1 was answered by , our top Math solution expert on 06/21/17, 07:45AM. The answer to “Suppose that a and b are integers. a ? 11 (mod 19), and b ? 3 (mod 19). Find the integer c with 0 ? c ? 18 such thata) c ? 13a (mod 19).________________b) c ? 8b (mod 19).________________c) c ? a - b (mod 19).________________d) c ? 7a + 3b (mod 19).________________e) c ? 2a2 + 3b2 (mod 19).________________f) c ? a3 + 4b3 (mod 19).” is broken down into a number of easy to follow steps, and 68 words. Since the solution to 14E from 4.1 chapter was answered, more than 409 students have viewed the full step-by-step answer. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. This full solution covers the following key subjects: mod, Find, integers, Integer, such. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

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Suppose that a and b are integers. a 11 (mod 19), and b 3