Introductions to Sets

Introductions to SetsUnit 15 > Lesson 1 of 14 Example 1: Kyesha was in math class with her friend Angie. She whispered to Angie that she had just bought a set of winter clothes. The outerwear collection includes a coat, a hat, a scarf, gloves, and boots. Their teacher, Mrs. Glosser, overheard the conversation and asked them: What is a set? Solution: Luckily for Kyesha and Angie, their classmate Eduardo had a math dictionary with him! He quickly looked up the word "set" and defined it for the class as shown below. A set is a collection of objects that have something in common or follow a rule. The objects in the set are called its elements. Set notation uses curly braces, with elements separated by commas. So the set of outwear for Kyesha would be listed as follows: A = {coat, hat, scarf, gloves, boots}, where A is the name of the set, and the braces indicate that the objects written between them belong to the set. Every object in a set is unique: The same object cannot be included in the set more than once. Let's look at some more examples of sets. Example 2: What is the set of all fingers? Solution: P = {thumb, index, middle, ring, little} Note that there are others names for these fingers: The index finger is commonly referred to as the pointer finger; the ring finger is also known as the fourth finger, and the little finger is often referred to as the pinky. Thus, we could have listed the set of fingers as: P = {thumb, pointer, middle, fourth, pinky} Example 3: What is the set of all even whole numbers between 0 and 10? Solution: Q = {2, 4, 6, 8} Note that the use of the word between means that the range of numbers given is not inclusive. As a result, the numbers 0 and 10 are not listed as elements in this set. Example 4: Eduardo was in art class when the teacher wrote this on the chalkboard: In fine arts, primary colors are sets of colors that can be combined to make a useful range of colors. Then she asked the class: What is the set of primary colors? Solution: Eduardo answered: red, blue and yellow. Angie answered: We can use set notation to list the set of all primary colors. Kyesha went to the chalkboard and wrote: X = {red, blue, yellow} The teacher said: Good work everyone. This is a nice combination of art and math! In examples 1 through 4, each set had a different number of elements, and each element within a set was unique. In these examples, certain conventions were used. The following conventions are used with sets:    Capital letters are used to denote sets. Lowercase letters are used to denote elements of sets. Curly braces { } denote a list of elements in a set. So for examples 1 through 4, we listed the sets as follows: 1. 2. 3. 4. A = {coat, hat, scarf, gloves, boots} P = {thumb, index, middle, ring, little} Q = {2, 4, 6, 8} X = {red, blue, yellow} These sets have been listed with roster notation. Roster notation is a list of elements, separated by commas, enclosed in curly braces. The curly braces are used to indicate that the elements written between them belong to that set. Let's look at some more examples of sets listed with roster notation. Example 5: Let R be the set of all vowels in the English alphabet. Solution: R = {a, e, i, o, u} Example 6: Let G be the set of all whole numbers less than ten. Solution: G = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} Example 7: Let T be the set of all days in a week. Solution: T = {Monday, Tuesday, Wednesday, Thursday, Friday} Example 8: Let X be the set of odd numbers less than 12. Solution: X = {1, 3, 5, 7, 9, 11} Example 9: Let Y be the set of all continents of the world. Solution: Y = {Asia, Africa, North America, South America, Antarctica, Europe, Australia} There are times when it is not practical to list all the elements of a set. In this case, it is better to describe the set. The rule that the elements follow can be given in the braces. For example,: R = {vowels} means Let R be the set of all vowels in the English alphabet. This is especially useful when working with large sets, as shown below. A = {types of triangles} G = {letters in the English alphabet} J = {prime numbers less than 100} M = {state capitals in the US} When describing a set, It is not necessary to list every element in that set. Thus, there are two methods for indicating a set of objects: 1) listing the elements and 2) describing the elements. We will distinguish between these two methods in examples 10 and 11 below. Example 10: Listing elements: What is the set of all letters in the English alphabet? D = {a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z} Describing elements: D = {letters in the English alphabet} Feedback to your answer is provided in the RESULTS BOX. five. 7. Select your answer by clicking on its button.Example 11: What is the set of all states in the Unites States? Solution: R = {all states in the US} In example 10. Summary: A set is a collection of objects that have something in common or follow a rule. queen. in example 11. so it is easier to describe its elements than to list them. 1. Which of the following is the set of all suits in a standard deck of playing cards? R = [ace. diamonds. four. setR has 50 elements. set D has 26 elements. 9} . Exercises Directions: Read each question below. two. 8. six. 2. We can define a set by listing its elements or by describing its elements. spades} T = {jokers} None of the above. Similarly. so it is easier to describe its elements. king] S = {hearts. Every object in a set is unique. eight. jack. 1. then choose a different button. rethink your answer. 3. Curly braces are used to indicate that the objects written between them belong to a set. RESULTS BOX: 2. 6. It is not necessary to list every object in the set. the rule that the objects follow can be given in the braces. ten. 4. nine. If you make a mistake. 5. three. clubs. The latter method is useful when working with large sets. seven. Instead. The objects in the set are called its elements. Which of the following is the set of odd whole numbers less than 10? C = {0. copper. carbon dioxide} Z = {liquids. silver} Y = {hydrogen. Jennifer listed the set of all letters in the word library as shown below. r. Australia} All of the above. 4. Pacific. gases. solids. Which of the following is the set of all oceans on earth? G = {Atlantic. 7. Africa. oxygen. Mississippi. Which of the following is the set of all types of matter? X = {iron. Nile. RESULTS BOX: 5. South America. a. 9} None of the above. Antarctic} E = {Amazon. 8} E = {1.D = {0. RESULTS BOX: 4. 3. What is wrong with this set? A = {l. r. RESULTS BOX: 3. y} . 6. aluminum. plasmas} None of the above. North America. Antarctica. Indian. nitrogen. 2. Arctic. b. i. gold. Europe. 5. nickel. Niagara} F = {Asia. Rio Grande. 4. We can also write 7 A. The objects in this set are not unique. Symbol Meaning is an element of is not an element of For example. 3. An object that belongs to a set is called an element (or a member) of that set. . which stands for 1 is an element of set A. 5}. They wrote about it on the chalkboard using set notation: P = {Kyesha. Glosser asked Kyesha. Angie and Eduardo to join the new math club. After school they signed up and became members.A capital letter is used to represent this set. Let's look at some more examples of this. she looked at the chalkboard and asked: What does that mean? Solution: Let P be the set of all members in the math club. as shown below. RESULTS BOX: Basic Set Notation Unit 15 > Lesson 2 of 14 Problem 1: Mrs. We use special notation to indicate whether or not an element belongs to a set. It uses commas. 2. we can write 1 A. Angie and Eduardo} When Angie's mother came to pick her up. which stands for 7 is not an element of set A. given the set A = {1. It uses curly braces. 4. 8. u} e w C = {1. 1. 7. 5. 9} 7 2 D = {-3. -1. 3. 10} 10 D = {English alphabet} m D = {English alphabet} X = {prime numbers less than 10} 9 A = {even numbers} 8 Summary: An object that belongs to a set is called an element (or a member) of that set. 6. 3} Determine if the given item is an element of the set. 8} Notation Meaning 2 5 A A B B C C 2 is an element of A 5 is not an element of A e is an element of B w is not an element of B 7 is an element of C 2 is not an element of C -2 is an element of D One-half is not an element of D B = {a. Set R = {2. 2. .Set A = {2. e. 6. 4. 8} Item 10 Is an element? S = {2. 0. 4. -2. i. 6. o. We use special notation to indicate whether or not an element belongs to a set: ( ). rethink your answer. gas. Select your answer by clicking on its button. If you make a mistake. then choose a different button. Which of the following is true for set R? R = {liquid. solid. Which of the following is true for set B? B = {US flag colors} red B . Which of the following is true for set G? G = {1. plasma} gas solid liquid R R R None of the above. 1. RESULTS BOX: 2. 9} 5 7 3 G G G All of the above.Exercises Directions: Read each question below. RESULTS BOX: 3. 7. 5. Feedback to your answer is provided in the RESULTS BOX. 3. lion. cougar.blue white B B All of the above. puma. Which of the following elements is not a member of set A? A = {states in the US} Guam Haiti Philippines All of the above. cheetah. ocelot} cougar bobcat puma tiger RESULTS BOX: 5. leopard. RESULTS BOX: 4. RESULTS BOX: Types of Sets Unit 15 > Lesson 3 . Which of the following elements is not a member of set X? X = {tiger. n. b. x. 1.. f. z} In example 2. 6. Describe this set using roster notation.. we used an ellipsis at the end of the list to indicate that the set goes on forever. Describe this set using roster notation. e. Describe this set using roster notation. It is not possible to explicitly list out all the elements of an infinite set. x. S = {a.} In example 3. t. o. 5. Let's look at some more examples of finite and infinite sets. d. g. . as shown in examples 1 and 2. i. u} Example 1: Solution: Example 2: Solution: Let S be the set of all letters in the English alphabet. S = {a. h.. 4. Set T is an infinite set. u. 3. So each of these sets is a finite set. c. v. there are 26 elements in set S. It would be easier to use a shortcut to list this set: Example 2: Solution: Let S be the set of all letters in the English alphabet. e. c. b.of 14 We learned how to write sets using roster notation. y. 2.. . Solution: T = {0. A finite set has a finitenumber of elements. FINITE SETS INFINITE SETS . Note that the number of elements in set R and set S is countable. q. o. s. R = {a. r. Let R be the set of all vowels in the English alphabet. m. i. l.. z} The three dots are called an ellipsis. j. An infinite set is a set with an infinite number of elements. w. p. We use an ellipsis in the middle of a set as a shortcut for listing many elements. y. k. Let's examine another type of set: Example 3: Let T be the set of all whole numbers. atom3.. 11} Description W = {even whole numbers} X = {atoms in the universe} Y = {prime numbers} Roster Notation W = {0. Empty (Null) Sets Description The set of dogs with sixteen legs. which goes on forever. Solution: D = {} We call a set with no elements the null or empty set.Description A = {whole numbers between 0 and 100} B = {primary colors} C = {prime numbers less than 12} Roster Notation A = {1.. . The set of computers that are both on and off. 3. . yellow} C = {2. So D = {} or D = Ø. blue. Example 4: Let D be the set of all weeks with 8 days. The set of triangles with 4 sides. 3. 2. 2. F = Ø Summary: An ellipsis is a shortcut used when listing sets with roster notation.. 5.} The ellipsis makes it easier to list both finite and infinite sets with roster notation. There are some sets that do not contain any elements at all.} Y = {2. 97. such as the set of whole numbers. 11. Notation X = {} Y = {} Z = {} D=Ø E=Ø The set of whole numbers that are odd and even. 6.. It is represented by the symbol { } or Ø . Let's look at some more examples of empty sets. as shown below... .. atom2. 5. 8. 98. 7. 99} B = {red. The set of bicycles with no wheels. A finite set has a countable number of elements: An infinite has an infinite number of elements. ....} X = {atom1. 7. We call a set with no elements the null or empty set. The set of months with 32 days. 3. 4. RESULTS BOX: 3. Select your answer by clicking on its button. gases.. then choose a different button. +3. . Which of the following sets are finite? {vowels} {days of the week} {primary colors} All of the above. -1. . RESULTS BOX: 2. rethink your answer. Feedback to your answer is provided in the RESULTS BOX.} empty finite infinite None of the above.. plasmas} empty finite infinite None of the above.Exercises Directions: Read each question below... -3. solids.. -2. +2. What type of set is H? H = {. 0. What type of set is G? G = {liquids. +1. 1. If you make a mistake. lion. Which of the following is an infinite set? {integers} {states in the US} {alphabet} None of the above. Which student used the correct notation? Student Notation Eduardo X = {red. cheetah. RESULTS BOX: Set Equality Unit 15 > Lesson 4 of 14 Problem 1: Mrs. She received two different answers from two different students as shown below. blue} . Glosser asked her class to write the set of primary colors using roster notation. yellow. puma. cougar. leopard. Which of the following is an empty (null) set? {tiger. RESULTS BOX: 5. ocelot} {cars with more than 20 doors} {prime numbers between 1 and 100} All of the above.RESULTS BOX: 4. Let's look at some more examples of set equality. 7} B = {3. e. 3. o. a} Examine these sets closely to confirm that they are equal. Answer: X=Y Since X and Y contain exactly the same number of elements. 5} Examine these sets closely to confirm that they are equal. and the elements in both are the same. i. and we write A = B. 1. The order in which the elements appear in the set is not important. Example 1: Solution: Are sets A and B equal? A = {1. Example 3: Are sets P and Q equal? . and we write X = Y. The equals sign (=) is used to show equality. we say that A is equal to B. o. 5.Angie Solution: Y = {blue. yellow} Both students used the correct notation. The sets from problem 1 are equal. red. e. 7. Example 2: Solution: Are sets X and Y equal? X = {a. we write X = Y. i. Answer: A=B Since A and B contain exactly the same number of elements. Remember that the order in which the elements appear in the set is not important. u} Y = {u. pears. 3. Jane. Joe. 8. i. 10} F = {John. and we write P ≠ Q. 8} K = {Jane. 6. 1} H = {o. 3. 2. 6. u} E = {2. a. 3. 0. we say that P is not equal to Q. pears} Q = {oranges. 4. apples} Examine these sets closely to confirm that they are not equal. Answer: P≠Q Since P and Q do not contain exactly the same elements. Answer: R=S Example 5: Which of the following sets are equal? C = {1. Example 4: Solution: Let R be the set of all whole numbers less than 5. e. 2. and let S = {4. 0. John} Answer: C = G and F = K . y.Solution: P = {apples. e. Are sets R and S equal? R = {whole numbers < 5} S = {4. u} J = {2. 2. 3} D = {a. bananas. o. oranges. Joe} G = {2. 1} Examine these sets closely to confirm that they are equal. 1}. 4. Feedback to your answer is provided in the RESULTS BOX. Which of the following sets is not equal to set H? H = {5.In example 5. 6. 6. 1. 2. Friday. Friday. 2. rethink your answer. Monday} X = {Tuesday. Wednesday. 2. Tuesday. then choose a different button. these sets are NOT equal: D ≠ H and E ≠ J. The order in which the elements appear in the set is not important. 4. 4. Thursday. Thursday. 1. 6} M = {3. 3} D = {1. and their elements are the same. If you make a mistake. Wednesday} All of the above. 5. RESULTS BOX: 2. Which of the following sets is equal to set P? P = {Monday. Tuesday. Exercises Directions: Read each question below. Saturday. 3. Select your answer by clicking on its button. 6} Q = {4. Friday} W = {Thursday. Can you name other sets that are not equal? Summary: Two sets are equal if they have the exact same number of elements. 2. Monday. 5. 7. 1. Saturday} Y = {Thursday. Friday. 1. 3} . 4. Wednesday. Sunday. 4. Which of the following statements is true? M is an infinite set. red}. RESULTS BOX: 4. . A is null and B has one element.None of the above. and let N = {even numbers < 10}. blue. A is a finite set and B is an infinite set. Let M = {0. 8. RESULTS BOX: 3. None of the above. M=N M≠N All of the above. RESULTS BOX: 5. and let B = Ø. 10}. Which of the following statements is true? X=Y Y=Ø X is an infinite set. 2. 6. Let X = {primary colors}. Which of the following statements is true? A is an infinite set and B is a finite set. Let A = {}. and let Y = {yellow. Put the elements in R.A = B. In aVenn diagram. usually circles or ovals. The elements of a set are labeled within the circle. Let's look at some examples. . We know from previous lessons that the following conventions are used with sets:    Capital letters are used to denote sets. RESULTS BOX: Venn Diagrams Unit 15 > Lesson 5 of 14 Until now. Lowercase letters are used to denote elements of sets. Label it G. Label it R. we have examined sets using set notation. Put the elements in G. first developed by John Venn in the 1880s. Analysis: Solution Draw a circle or oval. Another way to look at sets is with a visual tool called a Venn diagram. Draw a circle or oval. sets are represented by shapes. Example 1: Given set R is the set of counting numbers less than 7. Draw and label a Venn diagram to represent set G and indicate all elements in the set. Draw and label a Venn diagram to represent set R and indicate all elements in the set. Notation: R = {counting numbers < 7} Example 2: Analysis: Given set G is the set of primary colors. Curly braces { } denote a list of elements in a set. where the circles overlap. Example 4: Analysis: Let X = {1. 4. Draw a circle or oval. Put the elements in B. we will use a Venn diagram to find the intersection of two sets. . Notation: B = {vowels} In each example above. Draw and label a Venn diagram to represent set B and indicate all elements in the set. 2. Draw a picture of two overlapping circles.Solution: Notation: G = {primary colors} Example 3: Analysis: Solution: Given set B is the set of all vowels in the English alphabet. Label it B. Elements that are common to both sets will be placed in the middle part. 5}. First. Draw and label a Venn diagram to show the intersection of sets X and Y. we used a Venn diagram to represent a given set pictorially. We need to find the elements that are common in both sets. as we will see in the examples below. 3} and Let Y = {3. The intersection of two sets is all the elements they have in common. Venn diagrams are especially useful for showing relationships between sets. we look at all the elements in the two sets together. The union of two sets is the set obtained by combining the elements of each. It means "X or Y". This is what X and Y have in common. 3} and Let Y = {3. So X union Y is {1. we will find the union of two sets. The Venn Diagram in example 4 makes it easy to see that the number 3 is common to both sets. and is read as "X union Y". That is their intersection. Both circles have been shaded to show the union of these sets. So the intersection of Xand Y is 3. or in their intersection is in their union. So Intersection means "X and Y". Example 5: Analysis: Solution: Let X = {1. Let's Intersection written as Union . 2. 4. 5}. Explanation: Any element in X. The union of two sets is written as compare intersection and union. 4. 5}. In example 5 below. Y.Solution: Explanation: The circle on the left represents set X and the circle on the right represents set Y. 3. The intersection of X and Y is written as and is read as "X intersect Y". The shaded section in the middle is what they have in common. To find the union of two sets. Draw and label a Venn diagram to represent the union of these two sets. 2. Venn diagrams are especially useful for showing relationships between sets. Which of the following is represented by the Venn diagram below? {A} A = {odd numbers between 0 and 10} A = {even numbers between 0 and 10} None of the above. If you make a mistake. We will also learn more about intersection and union in this unit. 1. rethink your answer. Summary: We can use Venn diagrams to represent sets pictorially. We will explore this topic in more depth in the next few lessons.read as meaning of X intersect Y X and Y X union Y X or Y Look for the elements in common to both combine all elements The examples in this lesson included simple Venn diagrams. such as the intersection and union of overlapping sets. RESULTS BOX: . then choose a different button. Feedback to your answer is provided in the RESULTS BOX. Select your answer by clicking on its button. Exercises Directions: Read each question below. kings. clubs. queens.2. Which of the following is represented by the Venn diagram below? . aces} None of the above. Which of the following is represented by the Venn diagram below? {B} B = {hearts. diamonds. spades} B = {jacks. RESULTS BOX: 3. RESULTS BOX: 4. 7. 5. 10} Q = {6.P = {2. 15} None of the above. 11. 7} X = {2. 7. 6. 5. 3. 5. RESULTS BOX: . 8. 9} P∩Q All of the above. Which of the following is the correct roster notation for set X? X = {2. 3. 11} X = {2. 6. 3. 4. 5}.5. This is denoted by: Example 1: A Venn diagram for the relationship between these sets is shown to the right. 4} and B = {1. what is the relationship between these sets? We say that A is a subset of B. since every element of A is also in B. 4. RESULTS BOX: Subset s Unit 15 > Lesson 6 of 14 Given A = {1. Which of the following relationships is shown by the Venn diagram below? X∪Y X∩Y X=Y All of the above. 3. Answer: A is a subset of B. 3. 2. . Example 3: Given P = {1.Another way to define a subset is: A is a subset of B if every element of A is contained in B. 4. d}. e. For example. r. a. The statement "P is not a subset of Q" is denoted by: Note that these sets do have some elements in common. 5. 4} and Q = {2. e} and Y = {r. Answer: X is a subset of Y. Example 2: Given X = {a.This is denoted by: A Venn diagram for the relationship between these sets is shown to the right. 6}. 3. what is the relationship between these sets? We say that X is a subset of Y. The . since every element of X is also in Y. what is the relationship between these sets? We say that P is not a subset of Q since not every element of P is not contained in Q. Both definitions are demonstrated in the Venn diagram above. we can see that 1 Q. 3. Definition: For any two sets. 2. 2. 5. 4}.intersection of these sets is shown in the Venn diagram to the right. 3. then A = B. 1. 5} and B = {3. Symbol Meaning is a subset of is not a subset of Example 4: Analysis: Given A = {1. 4.. it is clear that: Answer: A and B are equivalent. Thus A and B are equivalent. if A B and B A.. . Example 5: Answer: List all subsets of the set C = {1. Subset D = {1} Comment List all possible combinations of elements. 2. 3}. Answer: P is not a subset of Q. Looking at the elements of these sets. what is the relationship between A and B? Recall that the order in which the elements appear in a set is not important. The notation for subsets is shown below. it can get a bit confusing. You may also be wondering: Is a set a subset of itself? The answer is yes: Any set contains itself as a subset. Do you see a pattern in the examples below? Example 6: List all subsets of the set R = Example 7: List all subsets of the set C = . This is because P and C are equivalent sets (P = C). This is denoted by: A A. then A is said to be a proper subset of B and it is denoted by A B. 2} is a proper subset of the set {1. the null set is a subset of every set. So the set {1.E = {2} F = {3} G = {1. 2. 3} N = {2. 2. 2} is a proper subset of the set {1. There are no elements in a null set. A subset that is smaller than the complete set is referred to as a proper subset. 2. every subset listed in example 5 is a proper subset of C. Some mathematicians use the symbol to denote a subset and the symbol to denote a proper subset. 3} P = {1. While it is important to point out the information above. 3} Ø one at a time one at a time two at a time two at a time two at a time three at a time The null set has no elements. with the definition for proper subsets as follows: If A B. The set {1. In example 5. 2. So let's think of subsets and proper subsets this way: Subsets and Proper Subsets The set {1. except P. you can see that G is a proper subset of C. so there can be no elements in the null set that aren't contained in the complete set. 3} because the element 3 is not in the first set. and A ≠ B. 3} is a not a proper subset of the set {1. 3}. Looking at example 5. you may be wondering why the null set is listed as a subset of C. In fact. 2} M = {1. 3}. 2. Therefore. 3} K = {1. y.{x. Answer: {1. 2. 4} S = {1. set R has three (3) elements and eight (8) subsets. z}. 4} O = {1. 2. . raise 2 to the nth power: That is: The number of subsets in set A is 2n . 3. y. To find the number of subsets of a set with n elements. If A B. z} J = {y. 2. where n is the number of elements in set A. 4} L = {2. This is denoted by A A. 2. z} Ø Answer: There are eight subsets of the set R = {x. 4} H = {1. 2. if A B and B A. 3} R = {2. 3. 4} Ø There are 16 subsets of the set C = {1. y} H = {x. where n is the number of elements in set A. z} K = {x. and A ≠ B. 4} N = {3. then A is said to be a proper subset of B and it is denoted by A B. For any two sets. 3. How many are there? Subsets D = {1} E = {2} F = {3} G = {4} M = {2. 2} Q = {1. 3. set C has four (4) elements and 16 subsets. Lesson Summary Subset Equivalent Sets Null set Sets and subsets Proper Subsets Number of Subsets A is a subset of B if every element of A is contained in B. 4} J = {1. 4}. In example 7. 4}. How many are there? Subsets D = {x} E = {y} F = {z} G = {x. then A = B. This is denoted by A B. y. The null set is a subset of every set. Any set contains itself as a subset. 3. In example 6. z}. 3} P = {1. The number of subsets in set A is 2n . 7. 11} B = {3. RESULTS BOX: 2. 7} C = {2. Which of the following is NOT a subset of set A? A = {2. rethink your answer.Exercises Directions: Read each question below. r} Y = {e. 11} All of the above. a. 5. RESULTS BOX: . RESULTS BOX: 3. 2. r. Feedback to your answer is provided in the RESULTS BOX. a} Z = {r. d} All of the above. 1. 3. 3. e} X = {e. 2. a. Which of the following statements is true? {vowels} {consonants} {vowels} {consonants} {vowels} {alphabet} None of the above. If you make a mistake. Which of the following is a subset of set G? G = {d. 7. e. Select your answer by clicking on its button. r. 5. then choose a different button. 9} D = {7. 3. then which of the following statements is true? R=S R has more elements than S. This relationship is shown in the Venn diagram below. 0.4. 2. If R = {whole numbers < 5} and S = {4. How many subsets will the set below have? T = {Monday. Tuesday. Wednesday. Thursday. 6} and B = {3. and that Venn diagrams can be used to illustrate both set relationships and logical relationships. Friday} 5 10 32 None of the above. 3. . RESULTS BOX: Universal Set Unit 15 > Lesson 7 of 14 In previous lessons. 1}. S is null. 5. what is the relationship between these sets? A and B have no elements in common. 2. 9}. RESULTS BOX: 5. None of the above. we learned that a set is a group of objects. Example 1: Given A = {1. 8. These sets do not overlap. denoted by capital All other sets are subsets of the universal set. draw a Venn diagram to represent these sets.) Therefore. 2. 2. 7. For example. then A and B are part of that set. In example 2. 3. then there is a relationship between them. and . or everything that is relevant to the problem you have. Definition: A Universal Set is the set of all elements under consideration. called the Universal Set. 7. 5. 2. (Each set is shaded with a different color to illustrate this. . 6. 9}. it is logical to assume that there is no relationship between these sets. 5. 4.Answer: A and B have no elements in common. 4. The procedure for creating a Venn . 9} is our larger set. 3. 9}. if we consider these sets as part of a larger set. Example 2: Given = {1. 6} and B = {3. A and B have no elements in common. In example 1. Note that subsets A and B do not overlap: These sets are disjoint. consider the single-digit numbers 1 through 9: If {1. A = {1. 5. 6. Answer: Think of a Universal set is the "big picture" It includes everything under consideration. 8. However. Thus A and B are each a subset of this larger set. Example 3: Given = {whole numbers less than 10}. subsets P and Q are overlapping. Example 4: Given = {whole numbers}. draw a Venn diagram to represent these sets. Draw circles within the rectangle to represent the subsets of the universe. draw a Venn diagram to represent these sets.diagram is as follows. . 1. P = {multiples of 3 less than 10} and Q = {even numbers less than 10}. 3. R = {primes numbers less than 12} and S = {even primes}. Draw a rectangle and label it U to represent the universal set. Let's look at some more examples. Write the remaining elements outside the circles but within the rectangle. Answer: In example 3. 2. Label the circles and write the relevant elements in each circle. draw a Venn diagram to represent these sets. . Accordingly. subsets X and Y do not overlap.Answer: In example 4. S is contained within R. we did not include any remaining whole numbers outside the circles and within the rectangle. Below is a word problem that you may find interesting. In addition. X = {dogs} and Y = {cats}. Example 5: Given = {animals}. since the set of whole numbers goes on forever. the universal set is infinite. This is due to the fact that the number 2 is the only even prime. Answer: In example 5. some students were selected for the school band. A universal set includes everything under consideration. Also included were examples in which one set was contained within the other. denoted by capital . the sets overlapped and in some they did not. In some examples. Band and Chorus are each a subset of the universal set. some were selected for the school chorus. In this lesson. Answer: In example 6. Band and Chorus are overlapping sets. Nathan. Select your answer by clicking on its button. . Raúl. Kyesha}. Lorrie. Kyesha. Chris. Feedback to your answer is provided in the RESULTS BOX. Shirley. Band = {Sam. If you make a mistake. some were selected for both. which is all the students in the class. we examined several examples of universal sets with Venn diagrams. Derek. then and . rethink your answer. Summary: A universal set is a set containing all elements of a problem under consideration. In addition. Exercises Directions: Read each question below. Robin. Lorrie. Derek} and Chorus = {Robin. or everything that is relevant to the problem you have. Given = {Sam.Example 6: In a class of 10 students. If the universal set contains sets A and B. Raúl. draw a Venn diagram to represent these sets. and the rest were selected for neither. Dana}. Derek. 1. Arctic. Antarctic}. then which of the following statements is true? M∩N=Ø All of the above. and Y = {Atlantic. If G = {-9. Antarctica. then which of the following could be the universal set? = {oceans} = {countries} = {world} All of the above. +2. then which of the following is the universal set? = {fractions} = {integers} = {irrationals} All of the above. -6} and H = {+8. South America. Europe. RESULTS BOX: 2. RESULTS BOX: . Africa. Indian. Australia}. Pacific. -7. RESULTS BOX: 3. M = {even numbers} and N = {odd numbers}. +4}.then choose a different button. If = {whole numbers}. If X = {Asia. -8. +7. North America. then which of the following sets does not overlap with A. 36}. RESULTS BOX: . 4. RESULTS BOX: 5. and is also is a part of ? B = {rectangles} C = {triangles} D = {parallelograms} None of the above. 25. 16. and is also a part of ? Q = {factors of 36} R = {multiples of 4} S = {even primes} None of the above. 9. If = {polygons} and A = {quadrilaterals). then which of the following sets overlaps with P.4. If = {whole numbers less than 40} and P = {1. 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