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Sets & Venn Diagrams

Math 4371: Stevie Vines, Keelin Wiley, Meredith Knox
by

Stevie Dodds

on 3 December 2012

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Transcript of Sets & Venn Diagrams

Lets Learn Sets and Venn Diagrams!! First, lets review the basics of sets! What Are Sets? Set Arrangement A set is a collection of items where order does not matter. We can Learn How Venn Diagrams Can Show a relationship between sets! Step one! The items within a set are called elements. Ex: Let set A be the numbers 1, 2, 3.

A={1, 2, 3}

So the numbers 1, 2, and 3 are elements of set A! What are Elements? Step Two! Let's take a look at different kinds of set relationships! Disjoint Sets... Subsets. Equal Sets. These 2 sets have no relationship with each other, which makes them "disjoint Sets". A={1,2,3,4} B={5,6,7,8} The elements in the smaller set of numbers is part of the bigger set of numbers, so the small set is a called a "subset" of the bigger one. B={2,4,6} A={1,2,3,4,5,6,7,8,9} subset of bigger set!!! If each set has identical things in it, the sets are "equal sets". This set is all whole numbers! C={1,2,3,4,5,6,7,8,9,10...} This set starts at 1 and you add one for each Element!! A={1,2,3,4,5,6,7,8,9,10...} There are different ways to find the "elements", but they will all be the same! Step Three! HERE... NOW HERE... AND NOW HERE. Step Four! Using Venn Diagrams to represent Sets! 5 10
15 What are Venn Diagrams? 2 6
8 11 A Venn diagram is a drawing where circular areas represent groups of items sharing or not sharing elements. 2 5
8 11 5 10 15 purple
red
blue green
orange Relationships of Sets in Venn Diagrams! 6 8
10 1 3 5
7 9 Elements in this section labeled A & B are elements that belong to both set A and set B. The area where the circles overlaps is called the "intersection" of the two sets. Step five! We can use Venn Diagrams to represent data as well! step six! We can also use Venn diagrams to solve word problems too!
Lets try one! = Sets can contain your favorite numbers, the days of the week, or even the names of your family and friends. Ex: Let set B be the shapes B={ , , } So the triangle, circle and square are elements of set B. The elements of a set are listed by roster! This means the elements are separated by commas and surrounded by curly braces! Empty Set If there are no elements in the set then the set is called an empty set. 5 A B C A Venn diagram consists of two or more circles each representing a set with a square representing the universal set. U 2
4 A B A & B Elements in the section labeled A are elements that are only in Set A. Elements in the section labeled B are elements that are only in Set B. This Venn diagram shows two sets A and B that overlap, which means there are some elements that exists in both A and B. The notation A or B represents the entire region covered by both sets A and B and the section where they overlap and is called the "union" of the two sets. A B A & B That would be everything in all three sections of the circles! Now lets try some examples! Let U (universal set)= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Let A= {2, 4, 6, 8, 10}
Let B= {1, 2, 3, 4, 5, 7, 9}
Represent the sets using a Venn Diagram. U A B Answer! What is the intersection of set A and B? A & B = {2, 4} What is the union of set A and B? (A or B) A or B= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} Let U= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15}
Let A= {2, 4, 6, 8, 10, 12, 14}
Let B= {3, 6, 9, 12, 15}
Let C= {4, 8, 12}
Represent the sets using a Venn Diagram. U A B C 12 6 4 8 2 10 14 3 9 15 Answer! What is the intersection of set A and C? A & C= {4, 8, 12} What is the intersection of A and B? A & B= {6, 12} What is the intersection of set A, B, and C? A & B & C= {12} What is the union of A and C? A or C= {2, 4, 6, 8, 10, 12, 14} U Skiing Soccer Swimming 17 11 20 18 14 2 12 15 What is the subset of A? Set C is a subset of set A
A= {2, 4, 6, 8, 10, 12, 14}
C= {4,8,12} How many students like Skiing or Soccer? Skiing or (union) soccer = {14 + 11 + 17+ 20+ 18+ 12} = {92} How many students like both skiing, soccer, and swimming Swimming, Soccer and Skiing (intersect)= {17} How many students only like Skiing? 24 dogs are in a kennel. 12 of the dogs are black, 6 of the dogs have short tails, and 15 of the dogs have long hair. There is only 1 dog that is black with a short tail and long hair. 2 of the dogs are black with short tails and do not have long hair. 2 of the dogs have short tails and long hair but are not black. 3 of the dogs have long hair and are black but do not have short tails. If all of the dogs in the kennel have at least one of the characteristics, how many dogs are only black? Black Short Tails Long Hair 24 Total All 3 1 Black & Short 2 Black & Long Long & Short 2 x 3 1 9 Elements in only A Elements in only B Elements in both A and B This will be elements in the A section, elements in the B section, and elements in the middle section! This will be elements in the middle section of the circles! What is the union of set A and B? A or B= {2, 3, 4, 6, 8, 9, 10, 12, 14, 15} Skiing only= {14} For this one, we will add all the numbers in the Skiing circle and all the numbers in the Soccer circle! In this Venn diagram, it shows the number of students who like Skiing, Soccer, and/0r Swimming! 1) First, we will fill in the middle section which is 1.

2) Fill in the sections where there are two characteristics.

3) To find the dogs with only one characteristics we have to add up the other numbers of the circle and subtract from the total number given of that particular characteristic.

Short Tails= 6- 2+ 2+ 1= 1
Long Hair= 15- 3+ 1+ 2= 9
Black Hair= 12- 3+ 2+ 1= 6

Now, we know 6 dogs only have black hair so x=6! Now you know more about Sets and Venn Diagrams! How many students do not like Skiing, Soccer, or Swimming? This would be what is left outside of all 3 circles! Neither Skiing, Soccer, or Swimming= {15} For this one, the intersection of A and C will be elements that are in both A and C so we can't forget to include the middle section because that number is in set A and C also! This is like the one above! For this one, we have to include everything in circle A and everything in circle C without repeating elements! This is like the one above! Remember from before when a smaller set has some of the same elements as a larger set then it is a subset of the larger set!
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