DM Slides - 26 July

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are from Noeither London or Manchester. They study either We do not know what the studentsGeography or Economics. Some of the students are in London study - some could studyfrom London and the rest study Economics Economics.Place ‘Yes’ if the conclusion does follow 51Place ‘No’ if the conclusion does not followAll of the Economic students are from Manchester.

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are fromeither London or Manchester. They study eitherGeography or Economics. Some of the students arefrom London and the rest study EconomicsPlace ‘Yes’ if the conclusion does followPlace ‘No’ if the conclusion does not followNone of the students from London studyEconomics. 52

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are from Noeither London or Manchester. They study either We do not know what the studentsGeography or Economics. Some of the students are in London studyfrom London and the rest study Economics 53Place ‘Yes’ if the conclusion does followPlace ‘No’ if the conclusion does not followNone of the students from London studyEconomics.

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are fromeither London or Manchester. They study eitherGeography or Economics. Some of the students arefrom London and the rest study EconomicsPlace ‘Yes’ if the conclusion does followPlace ‘No’ if the conclusion does not followSome of the Manchester students studyEconomics. 54

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are from Yeseither London or Manchester. They study either All of the Manchester studentsGeography or Economics. Some of the students are study Economics, so this is true.from London and the rest study Economics 55Place ‘Yes’ if the conclusion does followPlace ‘No’ if the conclusion does not followSome of the Manchester students studyEconomics.

Syllogisms 56Example 2: SyllogismsA group of friends at Cambridge university are fromeither London or Manchester. They study eitherGeography or Economics. Some of the students arefrom London and the rest study EconomicsPlace ‘Yes’ if the conclusion does followPlace ‘No’ if the conclusion does not followThere is a possibility that some students inLondon study Economics and some studyGeography

SyllogismsExample 2: SyllogismsA group of friends at Cambridge university are from Yeseither London or Manchester. They study either We do not know what the studentsGeography or Economics. Some of the students are in London study, so there is afrom London and the rest study Economics possibility this is true.Place ‘Yes’ if the conclusion does follow 57Place ‘No’ if the conclusion does not followThere is a possibility that some students inLondon study Economics and some studyGeography

Syllogisms: Example 3 58

SyllogismsExample 3: SyllogismsSome cupboards are trees. Some trees are leaves. All leaves are junglesPlace ‘Yes’ if the conclusion follows. Place ‘No if the conclusion does not follow.1. Some leaves are cupboards2. Some jungles are cupboards3. All trees are leaves4. Some trees are jungles 59

SyllogismsExample 3: SyllogismsSome cupboards are trees. Some treesare leaves. All leaves are junglesWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves 604. Some trees are jungles

SyllogismsExample 2: SyllogismsSome cupboards are trees. Some treesare leaves. All leaves are junglesWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves4. Some trees are jungles

SyllogismsExample 2: SyllogismsSome cupboards are trees. Some trees Some leaves are cupboardsare leaves. All leaves are jungles FALSEWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves4. Some trees are jungles

SyllogismsExample 2: SyllogismsSome cupboards are trees. Some trees Some jungles are cupboardsare leaves. All leaves are jungles FALSEWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves4. Some trees are jungles

SyllogismsExample 2: SyllogismsSome cupboards are trees. Some trees All trees are leavesare leaves. All leaves are jungles FALSEWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves4. Some trees are jungles

SyllogismsExample 2: SyllogismsSome cupboards are trees. Some trees Some trees are junglesare leaves. All leaves are jungles TRUEWhich of these conclusions are true?1. Some leaves are cupboards2. Some jungles are cupboards Cupboards Trees Leaves Jungles3. All trees are leaves4. Some trees are jungles

SyllogismsExample 3: SyllogismsSome cupboards are trees. Some trees areleaves. All leaves are jungles.Which of these conclusions are true? Cupboards Trees Leaves Jungles1. Some leaves are cupboards - FALSE2. Some jungles are cupboards - FALSE3. All trees are leaves - FALSE4. Some trees are jungles - TRUE

SyllogismsSummary1. Learn how to draw Venn Diagrams for each type of statement2. Always draw a Venn Diagram to summarise the information.3. Go through each statement one by one, using the Venn Diagram4. Do not use external knowledge - the statements won’t make sense most of the time. 67

Venn Diagrams I Lesson 4 68

Venn DiagramsTypes of Venn Diagram Question Text to Venn They give you text, and ask you to choose which of the 4 Venns fit the information best. Implicit Text to Venn They don’t explicitly ask you to draw a Venn, but doing so will help to solve the problem. Euler Diagrams They give you real-life categories, and you use your own knowledge to pick the right Venn. 69

Venn DiagramsTypes of Venn Diagram Question Venn to Text They give you a Venn diagram, and you have to choose the correct statement. Non-circular Venn Diagrams They give you a Venn diagram, made of non circular shapes, and you have to choose the correct statement. 70

Text to Venn Type 1 71

Venn DiagramsType 1 - Text to VennThey give you… a text scenarioThey ask you… to choose the correct Venn Diagram to fit the scenarioYou have to… use the information step by step to find the correct Venn Diagram 72

Venn DiagramsType 1 - Text to VennThey give you… a text scenarioThey ask you… to choose the correct Venn Diagram to fit the scenarioYou have to… use the information step by step to find the correct Venn Diagram “Which diagram best represents the information”Your are given a text scenario, and are asked to choose which of the four Venn diagrams most accurately represents the information. 73

Venn DiagramsExample 1: Venn DiagramsHilary is making 16 cakes for her tea party. She is putting toppings on them - sprinkles,smarties and cookies. 7 contain sprinkles, 3 contain all three toppings, 2 cakes contain 2toppings. The same number of cakes have sprinkles only as those that have only cookies.Which diagram best represents the information?A B. C. D. 74

Venn DiagramsStep 1 1. Check that they all add up to correct number Hilary is making 16 cakes Hilary is making 16 cakes for her tea party. She is putting When counting up the numbers for each Venn toppings on them - sprinkles, smarties and cookies. 7 diagram we find: contain sprinkles, 3 contain all three toppings, 2 cakes A - 16, B - 19, C - 16, D - 16 contain 2 toppings. The same number of cakes have sprinkles only as those that have only cookies. D. Which diagram best represents the information? A B. C. 75

Venn DiagramsStep 2 Use the information with the most categories first. The scenario has information involving: Hilary is making 16 cakes for her tea party. She is putting toppings on them - sprinkles, smarties and cookies. 7 3 categories contain sprinkles, 3 contain all three toppings, 2 cakes 2 categories contain 2 toppings. The same number of cakes have 1 category sprinkles only as those that have only cookies. So start with the information with 3 categories… Which diagram best represents the information? D. A B. C. 76

Venn DiagramsStep 2 2. “3 contain all three toppings” Hilary is making 16 cakes for her tea party. She is putting Make sure the diagram has 3 in the middle where toppings on them - sprinkles, smarties and cookies. 7 all circles intersect. contain sprinkles, 3 contain all three toppings, 2 cakes contain 2 toppings. The same number of cakes have This works for all remaining options here. So this sprinkles only as those that have only cookies. is no help.Which diagram best represents the information? C. D. A B. 77

Venn DiagramsStep 3 3. “2 cakes contain 2 toppings” Hilary is making 16 cakes for her tea party. She is putting The three regions involving only two overlapping toppings on them - sprinkles, smarties and cookies. 7 circles have to sum up to 2. contain sprinkles, 3 contain all three toppings, 2 cakes contain 2 toppings. The same number of cakes have This does not help us either as all four diagrams sprinkles only as those that have only cookies. satisfy this.Which diagram best represents the information? C. D. A B. 78

Venn DiagramsStep 4 4. “The same number have sprinkles as those that have cookies” Hilary is making 16 cakes for her tea party. She is putting toppings on them - sprinkles, smarties and cookies. 7 This means that at the centre of each circle there contain sprinkles, 3 contain all three toppings, 2 cakes should be two numbers that are the same. contain 2 toppings. The same number of cakes have sprinkles only as those that have only cookies. This doesn’t work for D, so knock it out.Which diagram best represents the information? C. D. A B. 79

Venn DiagramsStep 4 4. “The same number have sprinkles as those that have cookies” Hilary is making 16 cakes for her tea party. She is putting toppings on them - sprinkles, smarties and cookies. 7 This means that at the centre of each circle there contain sprinkles, 3 contain all three toppings, 2 cakes should be two numbers that are the same. contain 2 toppings. The same number of cakes have sprinkles only as those that have only cookies. This doesn’t work for D, so knock it out.Which diagram best represents the information?A B. C. D. 80

Venn DiagramsStep 5 5. “7 contain sprinkles” Hilary is making 16 cakes for her tea party. She is putting There has to be at least one enclosed region toppings on them - sprinkles, smarties and cookies. 7 adding up to 7. contain sprinkles, 3 contain all three toppings, 2 cakes contain 2 toppings. The same number of cakes have This is only true for A. sprinkles only as those that have only cookies.Which diagram best represents the information?A B. C. D. 81

Venn DiagramsStep 5 5. “7 contain sprinkles” Hilary is making 16 cakes for her tea party. She is putting There has to be at least one enclosed region toppings on them - sprinkles, smarties and cookies. 7 adding up to 7. contain sprinkles, 3 contain all three toppings, 2 cakes contain 2 toppings. The same number of cakes have This is only true for A. sprinkles only as those that have only cookies.Which diagram best represents the information?A B. C. D. 82

Venn DiagramsExample 1: Venn DiagramsHilary is making 16 cakes for her tea party. She is putting toppings on them - sprinkles,smarties and cookies. 7 contain just sprinkles, 3 contain all three toppings, 2 cakes contain 2toppings. The same number of cakes have sprinkles only as those that have only cookies.Which diagram best represents the information?A B. C. D.Correct Answer Too many Doesn’t add up to 7 83 numbers sprinkles 2 don’t contain the same number of toppings

Implicit Text to Venn Type 2 84

Venn DiagramsType 2 - Implicit Text to VennThey give you… a text scenarioThey ask you… to choose the correct statement or answer to fit the scenarioYou have to… draw a Venn Diagram even though they don’t tell you to 85

Venn DiagramsExampleThe local council were doing a survey how many people have gadgets. They surveyed100 people in total.• 7 people have TVs, tablets and mobiles • 33 people have TVs• 11 people have TVs and tablets • 30 people have mobiles• 8 people have tablets and mobiles. • 27 people have tablets• 26 people have TVs and mobilesHow many people had neither tablets nor mobile phones?A. 51 86B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 87have electronic gadgets. They surveyed 100 people in total. TVs Mobile Phones7 people have TVs, tablets and mobile phones11 people have TVs and tablets8 people have tablets and mobile phones.26 people have TVs and mobile phones33 people have TVs30 people have mobile phones27 people have tabletsHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 7have electronic gadgets. They surveyed 100 people in total. 887 people have TVs, tablets and mobile phones11 people have TVs and tablets TVs Mobile Phones8 people have tablets and mobile phones.26 people have TVs and mobile phones33 people have TVs30 people have mobile phones27 people have tabletsHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 11 8have electronic gadgets. They surveyed 100 people in total. 77 people have TVs, tablets and mobile phones 2611 people have TVs and tablets8 people have tablets and mobile phones. 8926 people have TVs and mobile phones33 people have TVs TVs Mobile Phones30 people have mobile phones27 people have tabletsHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 41have electronic gadgets. They surveyed 100 people in total. 7 197 people have TVs, tablets and mobile phones11 people have TVs and tablets 908 people have tablets and mobile phones.26 people have TVs and mobile phones TVs Mobile Phones33 people have TVs30 people have mobile phones27 people have tabletsHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 27have electronic gadgets. They surveyed 100 people in total. 41 307 people have TVs, tablets and mobile phones 711 people have TVs and tablets8 people have tablets and mobile phones. 33 1926 people have TVs and mobile phones33 people have TVs 9130 people have mobile phones27 people have tablets TVs Mobile PhonesHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 15have electronic gadgets. They surveyed 100 people in total. 41 37 people have TVs, tablets and mobile phones 711 people have TVs and tablets8 people have tablets and mobile phones. 2 1926 people have TVs and mobile phones33 people have TVs 9230 people have mobile phones27 people have tablets TVs Mobile PhonesHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 15have electronic gadgets. They surveyed 100 people in total. 41 37 people have TVs, tablets and mobile phones 711 people have TVs and tablets8 people have tablets and mobile phones. 2 1926 people have TVs and mobile phones33 people have TVs 9330 people have mobile phones27 people have tablets TVs Mobile Phones They surveyed 100 people in total. Number with at least one gadget is 52. So the number with no gadgets is 100 - 52 = 48

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 15have electronic gadgets. They surveyed 100 people in total. 41 37 people have TVs, tablets and mobile phones 711 people have TVs and tablets8 people have tablets and mobile phones. 2 1926 people have TVs and mobile phones33 people have TVs 9430 people have mobile phones27 people have tablets TVs Mobile PhonesHow many people had neither tablets or mobile phones?People that only had TVs = 3People that had no tables = 48People with neither tablets or mobile phones = 51

Venn Diagrams Step by Step Guide TabletsThe local council were doing a survey how many people 15have electronic gadgets. They surveyed 100 people in total. 41 37 people have TVs, tablets and mobile phones 711 people have TVs and tablets8 people have tablets and mobile phones. 2 1926 people have TVs and mobile phones33 people have TVs 9530 people have mobile phones27 people have tablets TVs Mobile PhonesHow many people had neither tablets or mobile phones?A. 51B. 55C. 45D. 40

Venn Diagrams II Lesson 5 96

Euler Diagrams Type 3 97

Venn DiagramsType 3 - Euler DiagramsThey give you… a list of items which are setsThey ask you… to choose the correct Venn Diagram to represent the group of itemsYou have to… group the items using your own knowledge, and then find the correct Venn 98

Venn DiagramsType 3 - Euler DiagramsThey give you… a list of items which are setsThey ask you… to choose the correct Venn Diagram to represent the group of itemsYou have to… group the items using your own knowledge, and then find the correct Venn A Venn diagram show all possible logical relationships between a series of sets. But an Euler diagram only shows relationships that exist in real world. 99

Venn DiagramsType 3 - Euler DiagramsA Venn diagram show all possible logical relationships between a series of sets.But an Euler diagram only shows relationships that exist in real world.Syllogisms do not make sense logically…e.g. Apples are Pizzas 100