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Solved: Prove that if A1, A2,…, An and B are sets, then

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

Solution for problem 42E 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 42E

Prove that if A1, A2, … ,An and B are sets, then

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Perception in Context Constancies and Contrast effects Perceptual Set Top­Down Processing vs. Bottom­Up Processing Bottom­up: Receptive Fields, Feature Detectors, and the Binding Problem Top­Down: Perceptual Set and Contextual Interpretation Empiricism and Associationism Pavlovian Conditioning Basic Methods and Results Acquisition ● The CR slowly grows as the animal experiences more and more pairings of the CS and US Extinction ● Undoing of a previously learned response so that the response is no longer produced Spontaneous Recovery ● The CS elicits the CR even though someone went through extinction and had a rest period Inhibition ● Theories Pavlov's Account Expectations, Blocking, & Surprise The Garcia Effect Biases Familiarity The Conditioning Model of Anxiety Disorders Biases in Fear Conditioning Social Transmission of Phobias Drug Conditioning Instrumental (Operant) Learning Instrumental Learning James and Willful Action ● Golfer knows what swing should look like ● Magic writing: unconsciously write on paper while blindfolded Comparative Psychology Thorndike ● Law of effect ● Cat in a puzzle box and hits lever and escapes for food Comparing Pavlovian and Instrumental Learning Skinner ● Sharp distinction between classical and instrumental conditioning ○ Instrumental responses are operants: operate on environment bringing about a change that leads to consequence ● Studied pigeons to aid in war effort Shaping ● You can shape & create target behavior with successive approximations Secondary Reinforcer ● Conditioned; money, grades, and tokens Immediate Reinforcement ● Occurs immediately after the desired or undesired behavior Successive Approximations ● Series of rewards that act as positive reinforcement towards final behavior Some Applications Clinical: Behavior Modification ● Patient can be rewarded for getting out of bed or moving around the room Creativity ● Dolphins or other entertainment animals Computer Assisted Instruction ● Used in mental institutions ● Dicky and his glasses Avoidance Learning ● Learn a behavior to avoid a stressful situation Basics ● Rat doesn’t wait for heat lamp(bad thing); automatically jumps on ledge Persistence ● Dog will jump over a barrier hundreds of times to avoid bad thing; never come in contact with the bad thing Punishment ● Learns to avoid by not doing something; kid avoids punishment by not running into the street Mnemonic Strategies Introduction: Working Memory and Long­Term Memory Mnemonics Brute Force Rehearsal ● Repeating ​doesn’t get things into memory ● Can only remeber 7 +­ 2 items with rehearsal(short term memory) Maintenance Rehearsal ● 7 +­ 2; short term memory; simply repeat list Elaborative Rehearsal ● Add meaning or rhythm to a set of info ● Professors phone number Interactive Imagery ● Create image that makes the objects interact; eagle and locomotive Method of Loci ● Find someplace you have well established in your head and place items there ​ asic Method "S" and Synesthesia ● “Human memories are remarkably alike. They just use them differently.” ● Gave digit span test: “S” repeated back 30 digits; filled in a 10x10 digits sheet forward, backward, and diagonally ● Synesthesia: experience info in every sense Rhythm and Rhyme ● Giving a rhythm or rhyme makes it easier to memorize Prospective Memory ● Know there is something in particular that you need to remember but don't always think about it; tie to retrieval cues that will be present at the time you need it Peg Words ● Peg words to number to create interactive images ● 1 bun, 2 shoe, 3 tree, 4 door, 5 hive Meaningful Surrogates ● Add personal meaning to words or phrases Substitution ● Substituting a word to help remember another word/number Chunking ● Think in terms of chunks rather than individual pieces of information Links ● Put in in personal environment to link it to your life Generation ● learned through generations Discovering Organization ● Discover the existing organization of the material Imposing Organization ● Create your own organization that works best for you Precise Elaboration ● Elaborating in a way that is more closely tied to the meaning of the words Intentional versus Unintentional Learning ● Learn more when you don’t mean to learn the material The Stage Model of Memory Processing Iconic Memory ● Remembering for a brief time after you can’t see anymore Working Memory Compared to Long­Term Store ● Working memory contains short term memory & systems that can operate on short term memory­> can get to long term memory through consolidation; Has a capacity of 7 +­ 2 at any time ● Long term memory information can be pulled back into short term memory Introspective Differences ○ How we perceive our own memories Capacity ● Working memory has a smaller capacity than long term memory Confusion Errors ● Short term confusion is sound, long term confusion is meaning Serial Position in Free Recall ● Remember better at the beginning and end of a data set than the middle Amnesia ● Often caused by alcoholism or damage to the brain; Alcohol causes decreased thiamine Anterograde(Korsakoff’s Syndrome) ● Person looks and acts normal but will have problems with certain topics; short term memory is normal but loses consolidation; cannot get new information into long term memory; memory does NOT recover; wipes out consolidation permanently Retrograde (Consolidation) ● Affects what’s happened in the past; may not remember before amnesia happened; electroconvulsive therapy wipes out short term memory and beginning stages of consolidation; greatest memory lost is the events right before the cause of the amnesia, can go back an entire day; temporarily wipes out consolidation The Memory Process Encoding ● Processing events and information in your brain Retrieval Retrieval Cues ● Cue that’s going to trigger a memory Hierarchical Search ● Search for something in memory but start further up the hierarchy State­Dependent Memory ● The state that you’re in becomes part of the retrieval cue Reconstruction ● Fill in memory with information that seems plausible Testing (Practicing Retrieval) ● Forgetting Loss of Retrieval Cues ● Lose cues when you move away Interference ● Learn something and subsequently learn something else Thinking, Problem­Solving, and Judgment Thinking as Covert Action Problem Solving Algorithms versus Heuristics ● Algorithm is strictly analytical; heuristics is an estimate based off previous knowledge; heuristic= rule of thumb Expertise Chunking ● above Heuristics ● Rule of thumb Depth of Processing Set ● A particular set of information that one relies on Rigidity Functional Fixedness ● When you think of things in terms of their normal function Heuristic­Based Errors in Reasoning Representativeness ● You decide a category that something belongs to by how representative that thing is by our notion of things that belong in said category Anchoring ● The first information you hear anchors your judgement and you adjust it based on the proceeding information Availability ● Estimate how frequently something is by how readily we can pull it up in our memory Language Syntax ● Arrangement of words to create a coherent sentence Semantics The "Dictionary" Model (Collins & Quillian) ● A good dictionary is carefully edited so people don’t get lost; words have a hierarchical organization Typicality ● Faster at categorizing things that are closer to the image; hear bird, think ro in Cognitive Development Approaches to Development: Nature vs Nurture Maturational ● You mature and grow; happens automatically Experiential ● People are products of their experiences Interactional ● Development involves both experience and maturation Piaget's Approach Basic Concepts Schemas ● Any organized action pattern ● Certain order for ordering food at a restaurant Assimilation and Accommodation ● Assimilation: Adding new information to a schema ● Accommodation: Change schema to fit new information Piegots 4 Major Periods Sensorimotor ● Lasts about 1 ½ years Sucking ● Newborn develop this skill within hours of birth ● Can get better at extracting milk Object permanence ● Knowing an object exists even though you can’t see it Pre­Operational ● Last until the child is 1­5 years old Failure to conserve ● Knowing there is the same amount of something, it’s just in a different container Egocentrism ● The child is unable to understand that people have different perspectives Concrete Operations ● 5­8 years old ● Learns conservation Formal Operations ● Social support ● Speeds up development Egocentric speech or thought development ● Children just think aloud rather than actually talking to each other Another approach: Child as Novice Meta­Memory ● Children have terrible short term memory ● Can only remember 3 +­ 1 ● Teaching rehearsal increased their short term memory for a period of time Counting One­on­One ● Assign a number name to an object at a rate of 1 per object Stable Order ● The certain order in which we count (1,2,3,4,5) Cardinality ● The last number is the last number assigned and the number of object in the whole group Order Irrelevance ● You get the same number of objects regardless of which order you count them Abstraction (applies to any set) ● Use counting cue to count other things such as people or animals Language Acquisition ➔ Glue words together around 1 year old First Language ● 2­22 year old people acquire 5­15 words a day ● Language must be acquired while young, often before puberty ● Children have a greater plasticity Telegraphic utterances ● Just enough words to carry the meaning of a message Overgeneralization ● Over applies a word; goed, went Critical Period ● Period in which information is most easily aquired Second Language ● An adult can learn a second language more easily than first language Animal Language/studies Early Attempts Clever Hans ● Horse could read arithmetic problems and correctly answer by stomping foot ● Would stop stomping when the crowd gasped ● Didn’t actually know arithmetic, but rather displayed sensitivity to social cues Trained Chimpanzees ● Raised chimps as best as possible to be human children ● The whole point was to see if chimps could develop human language Washoe and others ● Chimp raised in compound where people all used sign language ● Washoe could use 100+ signed words ● Chimps use words to stand for other things, but have no clear human language Birds ● Song birds pick up songs from their parents while they are young Emotional Development Attachment Freud: “Cupboard Love” ● Whoever provides food will become the center of the child's attachment Animal Models Imprinting (Lorenz) ● Experimented with birds ● animals form a strong attachment and will follow for a long time the first moving object that they encounter ● Got birds attached to objects such as a tractor Contact Comfort (Harlow) ● Harlow’s monkeys ● Orphan monkey would cling to artificial mothers ● Monkeys would cling to terry cloth mother and only feed from the wire mother ● Motherless monkeys were terrible mothers themselves Childhood Attachment Loss (Bowlby) ● Studied children who were separated from their parents ● Children form intense bonds around 6 months of age Protest ● Child who has been attached and separate is very unhappy; cries, kicks, hits; lasts days to weeks Despair ● Child becomes inactive; whimpers quietly rather than kicking ● Subject to certain kinds of infections and suffer higher rates of mortality Detachment ● Children form attachment tonew​ primary care takers The Uses of Attachment Sexuality Development ● Rearing can affect sexual development Strange Situations Social/Emotional Learning of Fears ● Anxieties and fears can be learned ● Visual cliff: babies learn to not crawl off the “cliff” ○ Babies got cues that it was dangerous by watching their mothers Cohort Effects: Reasons to be Skeptical Agoraphobia: fear of going into public places John B. Watson argued there is no such thing as conscious though Children love their mothers bc they have working breasts

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Chapter 5.1, Problem 42E is Solved
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Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

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Solved: Prove that if A1, A2,…, An and B are sets, then