Sets and Venn Diagrams PDF

MANUEL A. BELANGO, DME  SET The collection of well-defined distinct objects. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word ‘distinct’ means that the objects of a...

MANUEL A. BELANGO, DME  SET The collection of well-defined distinct objects. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word ‘distinct’ means that the objects of a set must be all different. 1.The collection of students in BSED 1- Math class whose weight exceeds 35 kg 2. The collection of all the intelligent students in BSED 1-Science class Exercises: Write S if the given group or collection is a set and NS if it is not. Write your answer on a piece of paper. 1. collection of students in your MMW class whose surname starts with letter A 2. group of good looking male actors 3. collection of great people of the world 4. collection of beautiful flowers 5. collection of distinct letter of the word “university” 6. group of cities in the province of Cagayan 7. numbers greater than 5 but less than 12 8. group of enjoyable subjects in high school 9. group of students in our class who wear mask 10.group of reputable schools in Tuguegarao City Elements of Set: The different objects that form a set are called the elements of a set. The elements of the set are written in any order and are not repeated. Elements are denoted by small letters. Notation of a Set: A set is usually denoted by capital letters and elements are denoted by small letters If x is an element of set A, then we say x ϵ A. [x belongs to A] If x is not an element of set A, then we say x ∉ A. [x does not belong to A] For example: The collection of vowels in the English alphabet. Solution: Let us denote the set by M, then M = [a, e, i, o, u]. We say a ∈ M, e ∈ M, i ∈ M, o ∈ M and u ∈ M. Also, we can say b ∉ M, c ∉ M, d ∉ M, etc.  Representation of a Set In representation of a set the following methods are commonly used: A. Roster or listing method B. Rule method A. Roster form or listing method - elements of the set are listed within the pair of braces { } and are separated by commas. For example: 1. Let N denote the set of first five natural numbers. Therefore, N = {1, 2, 3, 4, 5} → Roster Form 2. The set of all vowels of the English alphabet. Therefore, V = {a, e, i, o, u} → Roster Form 3. The set of all odd numbers less than 9. Therefore, X = {1, 3, 5, 7} → Roster Form 4. The set of all natural number which divide 12. Therefore, Y = {1, 2, 3, 4, 6, 12} → Roster Form  5. The set of all letters in the word Mathematics. Therefore, Z = {m, a, t, h, e, i, c, s} → Roster Form 6. W is the set of last four months of the year. Therefore, W = {September, October, November, December} → Roster Form B. Rule Method - a rule, or the formula or the statement is written within the pair of brackets so that the set is well defined. - all the elements of the set, must possess a single property to become the member of that set. - sometimes, it is written in set-builder notation A = {x I x is….} For example: 1. P = {1, 2, 3, 4, 5, 6, 7, 8} P = {set of counting numbers less than 9} the set P in set-builder form is written as : P = {x | x is a counting number less than 9} For example: 2. G = {3, 6, 9, 12, 15} G = { set of multiples of 3 greater than 2 but less than 16} For example: 3. L = {2, 3, 5, 7} L = { set of prime numbers less than 8} For example: 4, M = {4, 5, 6, 7,….} M = { set of counting numbers greater than 3} For example: 5. N = { red, yellow, blue, white} N = {set of colors of the Philippine flag} A. Write each of the following sets in the Roster form: 1. The set of first five natural numbers. 2. The set of whole numbers less than 7. 3. The set of all positive odd divisors of 12. 4. The set of whole numbers less than 24 and divisible by 4. 5. The set of integers greater than -3 and less than 5. B. Write each of the following sets in the rule method: 1. {8, 10, 12, 14, 16} 2. {5, 10, 15, 20, 25, 30} 3. {1, 4, 9, 16, 25, 36} 4. {10, 100, 1000, 10000, 100000} 5. {b, c, d, f, g} Kinds of Sets 1. Empty Set or Null Set:  A set which does not contain any element is called an empty set, or the null set or the void set and it is denoted by ∅ and is read as phi.  In roster form, ∅ is denoted by { }.  An empty set is a finite set, since the number of elements in an empty set is finite. Examples: 1. The set of whole numbers less than 0. 2. B = {x I x is a composite number less than 4}. 2. Singleton Set:  A set which contains only one element is called a singleton set. Example: 1. A = {x I x is neither prime nor composite} 2. B = {x I x is a even prime number} 3. Finite Set:  A set which contains a definite number of elements is called a finite set.  Empty set is also called a finite set. Example: 1. The set of all colors in the rainbow. 2. P = {2, 3, 5, 7, 11, 13, 17,...... 97} 4. Infinite Set:  The set whose elements cannot be listed, i.e., set containing never-ending elements is called an infinite set. Example: 1. Set of all points in a plane 2. C = { x I x is a prime number} Cardinal Number of a Set:  The number of distinct elements in a given set A is called the cardinal number of A.  It is denoted by n(A). Example: 1. A = {1, 2, 3, 4} Therefore, n(A) = 4 2. B = set of letters in the word “ALGEBRA” Therefore, n(B) = 6 5. Equivalent Sets:  Two sets A and B are said to be equivalent if their cardinal number is same, i.e., n(A) = n(B). The symbol for denoting an equivalent set is ‘↔’. Example: A = {1, 2, 3} Here n(A) = 3 B = {p, q, r} Here n(B) = 3 Therefore, A ↔ B 6. Equal sets:  Two sets A and B are said to be equal if they contain the same elements. Every element of A is an element of B and every element of B is an element of A. Example: A = {p, q, r, s} B = {p, s, r, q} Therefore, A = B 7. Disjoint Sets  Two sets A and B are said to be disjoint, if they do not have any element in common. Example: A = {x : x is a prime number} B = {x : x is a composite number}. Clearly, A and B do not have any element in common and are disjoint sets. 8. Overlapping sets:  Two sets A and B are said to be overlapping if they contain at least one element in common. Example: A = {a, b, c, d} B = {a, e, i, o, u} 9. Subset:  If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A ⊆ B or B ⊇ A  The symbol ⊆ stands for ‘is a subset of’ or ‘is contained in’  Every set is a subset of itself, i.e., A ⊆ A Empty set is a subset of every set. A ⊆ B means A is a subset of B or A is contained in B. Null set or ∅ is a subset of every set. For example: 1. Let A = {2, 4, 6} B = {6, 4, 8, 2} Here A is a subset of B. Since, all the elements of set A are contained in set B. But B is not the subset of A. Since, all the elements of set B are not contained in set A. Note: If A ⊆ B and B ⊆ A, then A = B, i.e., they are equal sets.  Proper Subset: If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B. The symbol ‘⊂’ is used to denote proper subset. Symbolically, we write A ⊂ B. For example: 1. A = {1, 2, 3, 4} Here n(A) = 4 B = {1, 2, 3, 4, 5} Here n(B) = 5 We observe that, all the elements of A are present in B but the element ‘5’ of B is not present in A. So, we say that A is a proper subset of B. Symbolically, we write it as A ⊂ B 10. Power Set:  The collection of all subsets of set A is called the power set of A.  It is denoted by P(A). In P(A), every element is a set. Example: If A = {p, q} ,then P(A) = {∅, {p}, {q}, {p, q}} 11. Universal Set  A set which contains all the elements of other given sets is called a universal set.  The symbol for denoting a universal set is ∪. Example: 1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7} then U = {1, 2, 3, 4, 5, 7} 2. If A = {a, b, c} B = {d, e} C = {f, g, h, i} then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.  Operations on Sets When two or more sets combine together to form one set under the given conditions, then operations on sets are carried out.  Union of Sets  Union of two given sets is the smallest set which contains all the elements of both the sets.  To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated. Union of Sets Union of Sets  Intersection of Sets  Intersection of two given sets is the largest set which contains all the elements that are common to both the sets.  To find the intersection of two given sets A and B is a set which consists of all the elements which are common to both A and B. Intersection of Sets Intersection of Sets Example: Let set A = {3, 6, 9, 12} and set B = {3, 5, 7, 9} In this two sets, the elements 3 and 9 are common. Therefore, A ∩ B = {3, 9} Solved examples to find intersection of two given sets: 1. If A = {2, 4, 6, 8, 10} and B = {1, 3, 8, 4, 6}. Find intersection of two set A and B. Solution: A ∩ B = {4, 6, 8} Therefore, 4, 6 and 8 are the common elements in both the sets. 2. If X = {a, b, c} and Y = {ф}. Find intersection of two given sets X and Y. Solution: X ∩ Y = { } 3. If set A = {4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15} and set C = {1, 4, 7, 10, 13}. Find: a. A ∩ B b. B ∩ C c. A ∩ C  Difference of Two Sets  If A and B are two sets, then their difference is given by A - B or B - A.  A - B means elements of A which are not the elements of B.  A - B = {x : x ∈ A, and x ∉ B}  B - A = {x : x ∈ B, and x ∉ A} Difference of Two Sets Example: 1. If A = {2, 3, 4} and B = {4, 5, 6} Solution: A - B = {2, 3} B - A = {5, 6} 2. Let A = {a, b, c, d, e, f} and B = {b, d, f, g}. Find the difference between the two sets: (i) A and B (ii) B and A 3. Given three sets P, Q and R such that: P = {x I x is a natural number between 10 and 16}, Q = {y I y is a even number from 8 and 17} and R = {7, 9, 11, 14, 18, 20} Find: a. P - Q b. Q - R c. R - P d. Q – P Complement of a Set Complement of a Set Example: 1. If U = {0, 1, 2, 3, 4, 5, 6} and A = {0, 4, 6} find A'. Solution: We observe that 0, 1, 3, 5 are the only elements of U which do not belong to A. Therefore, A' = {0, 1, 3, 5 } 2. Let U = The set of letters in the English alphabet. A = The set of consonants in the English alphabet then A' = The set of vowels in the English alphabet. From the adjoining Venn diagram, find the following sets. 1. A' 2. B' 3. C' 4. C - A 5. B - C 6. A - B 7. A ∪ B 8. B ∪ C 9. A ∩ C 10. B ∩ C 11. (B ∪ C)' 12. (A ∩ B)' 13. (A ∪ B) ∩ C  Venn Diagrams  Pictorial representations of sets represented by closed figures are called set diagrams or Venn diagrams.  Venn diagrams are used to illustrate various operations like union, intersection and difference.  We can express the relationship among sets through this in a more significant way.  Venn diagrams are useful in solving simple logical problems. Mathematician John Venn introduced the concept of representing the sets pictorially by means of closed geometrical figures called Venn diagrams. In Venn diagrams, the Universal Set U is represented by a rectangle and all other sets under consideration by circles within the rectangle. VENN DIAGRAMS IN DIFFERENT SITUATIONS If a set A is a subset of set B, then the circle representing set A is drawn inside the circle representing set B.  If set A and set B have some elements in common, then to represent them, we draw two circles which are overlapping.  If set A and set B are disjoint, then they are represented by two non-intersecting circles. Example 1 If U = {1,2,3,4,5,6,7,8,9,10} A = {1,3,5,7,9,10} B = {3,4,5,7,8} AB A B 1 4 6 3 9 5 7 8 10 2 Example 2 If U = {1,2,3,4,5,6,7,8,9,10} A = {2,4,6,8,10} B = {4,6,8} A 2 5 1 B 10 4 6 7 8 3 9 Example 3 If U = {1,2,3,4,5,6,7,8,9,10} A = {2,4,6,8,10} B = {1,3,5} A B 2 1 6 10 3 4 5 8 7 9 Example 4 If U = {1,2,3,4,5,6,7,8,9,10} A = {2,4,5,8,10} B = {4,6,9,10} C = {1,4,6,} A B 2 5 10 9 8 AB C 4 6 1 3 7 C 1. In a class of 50 students, 18 take English, 26 take Math, and 2 take both English and Math. How many students in the class are enrolled in a. English only b. Math only c. neither English nor Math? 2. In a class, there are 13 students who play volleyball, 19 students who play basketball, 8 students who play basketball and volleyball, and 7 students who do not play basketball or volleyball. How many students are there in the class? 3. In a class there are 30 students. 21 students like GMA. 16 students like ABS-CBN. 6 students don't like GMA or ABS-CBN. How many students like a. both GMA and ABS-CBN? b. GMA or ABS-CBN? c. GMA but not ABS-CBN? d. ABS-CBN but not GMA? e. like only one of the networks? 4. There are 45 farmers in a barangay. Twenty farmers planted rice, 28 farmers planted corn, 20 farmers planted vegetables, 15 farmers planted rice and corn, 10 farmers planted rice and vegetables, 9 planted corn and vegetables, and 5 farmers planted all three crops. How many farmers planted: a. rice only? h. rice or vegetable? b. corn only? i. corn or vegetable? c. vegetable only? j. rice or corn but not vegetables? d. vegetables but not corn? k. exactly two crops? e. corn but not rice? l. neither of the three crops? f. rice but not vegetables? g. rice or corn? 5. In a hospital, the patients had the following ailments: 45 had malaria, 45 had hypertension, and 45 had peptic ulcer. 25 had malaria and hypertension, 20 had malaria and peptic ulcer, 10 had hypertension and peptic ulcer while 5 had all the ailments. How many patients are sick of a. malaria only? b. malaria but not hypertension? c. hypertension and peptic ulcer but not malaria? d. malaria or hypertension but not peptic ulcer? e. neither malaria nor peptic ulcer? 6. A group of 62 students were surveyed, and it was found that each of the students surveyed liked at least one of the following three fruits: apples, bananas, and oranges. 34 liked apples. 30 liked bananas. 33 liked oranges. 11 liked apples and bananas. 15 liked bananas and oranges. 17 liked apples and oranges. 19 liked exactly two of the following fruits: apples, bananas, and oranges a. How many students liked apples, but not bananas or oranges? b. How many students liked oranges, but not bananas or apples? c. How many students liked all of the following three fruits: apples, bananas, and oranges? d. How many students liked apples and oranges, but not bananas?  7. A group of 70 people were surveyed, and it was found that each of them surveyed liked at least one of the following three pets: dogs, cats, and bird. 34 liked dogs. 30 liked cats. 33 liked bird. 11 liked dogs and cats. 15 liked cats and bird. 17 liked dogs and bird. 9 liked exactly three of the following pets: dogs, cats, and bird How many people liked a. dog only f. cat or bird b. cat only g. dog but not cat c. bird only h. bird but not dogs d. dog or cat i. dogs and bird, but not cats e. dog or bird j. exactly two of the pets