Learning objectives To introduce the basics of theory of sets and some of its applications Sets and their representation Cardinality of set Empty set Unit set Finite and infinite sets Equal and equivalent sets Subsets Power sets

Contents……..:

Contents…….. Universal set Venn diagram Operations of sets Union and Intersection of sets Disjoint sets Difference of sets Practical problems based on sets Tips and tricks

Sets :

Sets A set is a well-defined collection of objects. Comprises of any physical or mathematical objects. The objects are called its elements or members. For example: A set can be a pack of cards, states of India, types of geometrical shapes, numbers and so on. wwwbiotecnika.org

History:

History Founder of modern set theory “A set is a many that allows itself to be thought as a one” ……. Georg Cantor Georg Cantor (1845-1918) wwwbiotecnika.org

Why study sets? :

Why study sets? Fundamental property of mathematics Sets: Powerful building blocks of mathematics wwwbiotecnika.org

Uses of set theory:

Uses of set theory Defining and developing concepts of Relations and functions, Boolean algebra, Calculus , Geometry , and Topology . wwwbiotecnika.org

Applications of sets:

Applications of sets wwwbiotecnika.org

Sets and their representations……..:

Sets and their representations…….. Notation A ={ a, b, c, d, e} Elements/ objects denoted by small letters Sets is denoted by Capital letters If ‘a’ is an element of a set A , We say that “a belongs to A”, written as “a ∈ A”. If ‘f’ is not an element of a set A, We say “f does not belong to A ”, written as “f ∉ A” wwwbiotecnika.org Curly braces

Sets: Two methods of representations:

Sets: Two methods of representations Rooster form Set builder form Lists all the members Describes how the set is created Members s eparated by commas and enclosed in curly braces Members of the set denoted by using any symbol, followed by a colon “ : ” t hen common property of the elements and enclosed in curly braces. Examples: Examples: A = {a, e, i, o, u} A = { x: x is a vowel in English alphabet} P = { 2, 3, 5, 7} P = { x: x is a prime no. 1< x < 10} wwwbiotecnika.org

Cardinality of set:

The number of elements of a finite set is a natural number (a non-negative integer ). It is denoted by n(A ) or │A│. Cardinality of set For example, A = set of letters in the word ALGEBRA A = {A, L, G, E, B, R} Therefore, Cardinality of the set A= n(A) = 6 wwwbiotecnika.org

Empty set:

Empty set It is the unique set which doesn’t contain any elements. Its size and cardinality is zero . Empty set = Null set = void set. Common notations for the empty set are "{}" and "∅". For example: E= {Elephants with wings}. T = {x: x is a triangle with 4 sides} wwwbiotecnika.org

Unit/Singleton set :

Unit/Singleton set A set with only one element. For example: A = {x : x is neither prime nor composite} It is a unit set containing one element, i.e., 1 wwwbiotecnika.org

Finite sets:

Finite sets Contains a countable number of elements. Definite starting and ending points . Dots in the middle. For example: A= {2,4,……10, 12} is a finite set with 6 elements . F ={ No. of fingers in hand} D = { No. of days in a week} Set F Set D wwwbiotecnika.org

Infinite set:

Infinite set No. of elements cannot be counted or determined. Cardinality of the set = ∞ Dots at the end or beginning. For example: I = The set of all integers {…..-3,-2,-1, 0, 1, 2, 3 ,….} S ={ No. of stars in the sky} L = {Set of animals living in earth} Set S Set L wwwbiotecnika.org

Equal sets:

Equal sets Two sets A and B are equal if they contain the same elements. Every element of A is an element of B. E very element of B is an element of A. Otherwise , the sets are said to be unequal and we write A ≠ B For example: X = {The set of letters in dealer} = {D, E, A, L, R} Y = {The set of letters in leader} = {L, E, A, D, R} Z= {The set of letters in Biology}= {B, I, O, L, G,Y } Therefore, X = Y≠ Z A = B wwwbiotecnika.org

Equivalent sets:

Equivalent sets Same cardinal number of both the sets, n(A ) = n(B). The symbol is ‘↔’. For example: X = { 1, 2, 3}, n(X) = 3 Y = {e, f, g}, n(Y ) = 3 ∴ X ↔ Y L‘↔’ S A B C D A ↔ B ↔ C ↔ D wwwbiotecnika.org

Subsets:

Subsets Portion of a set where every element of A is also an element of B. Expressed as A ⊂ B . If A is not a subset of B, then A ⊄ B. For example : Natural numbers ⊂ whole numbers ⊂ Integers ⊂ Rational numbers ⊂ Real numbers Irrational numbers ⊄ Rational numbers wwwbiotecnika.org

Subsets in life:

Subsets in life Biosphere ⊂ Earth Organelles ⊂ Cell wwwbiotecnika.org

Power sets:

Power sets The power set is the collection of all subsets of that particular set. When cardinality of original set n (A) = n, Then Power Set, noted as |P(S)| will have 2 n members . For example: S = { a,b,c }, n (S) = 3, ∴ P(S ) = { {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c }} = 2 3 = 8 B ={Set of Blood group types} = { O + , A, B} P(B) ={ {O - }, {A}, {B}, { O + }, {A O + }, {B O + }, {A B}, {ABO + }} wwwbiotecnika.org

Universal set:

Universal set It is a set which contains all objects, including itself All the sets under consideration are subsets of a larger set. Denoted commonly as U, X or ξ V= {a, e, i, o, u} C= {b, c, d, f, g, h, i, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z } ξ For example : U = ξ = {Set of English alphabets}= { V : Vowels in alphabets} + {C : Consonants in alphabets} wwwbiotecnika.org

Venn diagrams:

Venn diagrams It is a geometric representation illustrating the relationships between and among sets. Named after John Venn (1834-1883), an English logician. The sets are represented usually by closed circles within a rectangle (universal set). The objects of the sets are written in their respective circles. Venn diagram showing multiple disciplines of science wwwbiotecnika.org

Operations of sets:

Operations of sets Four fundamental operations for constructing new sets from given sets: Union, Intersection , Difference and Complementation . wwwbiotecnika.org

Union and intersection of sets:

Union and intersection of sets A U B = All yellow, green, blue A ∩ B = Green A B Union of sets Intersection of sets Union of two sets A and B is the set of elements which are in A, in B, or in both A and B. Intersection of sets A and B is the set of all elements common to both A and B. A U B = { x: x ∈ A or x ∈ B}. A ∩ B = {x : x ∈ A and x ∈ B}. Denoted by ‘U’ Denoted as ‘∩’ wwwbiotecnika.org

Properties of Union and Intersection :

Properties of Union and Intersection Union of sets Intersection of sets A ∪ B = B ∪ A (Commutative law) A ∩ B = B ∩ A (Commutative law) A ∪ φ = A (Law of identity element, φ is the identity of ∪ φ ∩ A = φ(Law of φ) A ∪ A = A (Idempotent law) A ∩ A = A (Idempotent law) U ∪ A = U (Law of U) U ∩ A = A (Law of U) wwwbiotecnika.org

Properties on union and intersection of sets:

Properties on union and intersection of sets Let A, B and C be finite sets. Sets A – B, A ∩ B and B – A are disjoint, then n ( A ∪ B ) = n ( A ) + n ( B ) – n ( A ∩ B ) (A ∪ B ) ∪ C = A ∪ ( B ∪ C) (Associative law) ( A ∩ B ) ∩ C = A ∩ ( B ∩ C ) (Associative law) n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ) (Distributive law ) wwwbiotecnika.org

Disjoint sets:

Disjoint sets If A ∩ B = ∅, then A and B are said to be disjoint a b c d Jan Feb Mar April May June Set A = {a, b, c, d} Set B= { Months in half a year} Nothing similar among A and B Set H = {Horses} Set B= { Pencils in a stand} Nothing similar among H and B wwwbiotecnika.org

Difference of sets:

Difference of sets It is set of elements which belong to A but not to B. A – B = { x : x ∈ A and x ∉ B } A B A - B = { } wwwbiotecnika.org

Complement of sets:

Complement of sets Assuming U is the universal set and A, a subset of U. Complement of set A refers to all the objects of U which are not in (that is, objects outside of) A. Denoted as A ′ with respect to U. A ′ = {x : x ∈ U and x ∉ A }. A A Not A Set Complement wwwbiotecnika.org

Properties of complement sets:

Properties of complement sets A ′ = U – A. Complement laws: (i) A ∪ A′ = U (ii) A ∩ A′ = φ De Morgan’s law: (i) (A ∪ B)´ = A′ ∩ B′ (ii) (A ∩ B )′ = A′ ∪ B′ Law of double complementation: (A′ )′ = A Laws of empty set and universal set φ′ = U and U′ = φ A U A ′ wwwbiotecnika.org

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Practical problems based on sets:

Practical problems based on sets Q1. In a college, there are 30 teachers who teach mathematics or biology. Of these, 17 teach mathematics and 5 teach both biology and mathematics. How many teach Biology? 12 8 6 4 Answer: b) wwwbiotecnika.org

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Q2. In a room of 60 men, 30 like MS Dhoni , the cricketer and 23 like Ronaldo , the foot baller. Each man in the room likes at least one of the sportsmen. How many men like both sportsmen? 5 10 12 7 Answer d) wwwbiotecnika.org

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Q3 . In a class, 65% of the students enrolled for pottery and 50% enrolled for gardening. If 30% of the students enrolled for both pottery and gardening, what % of the students of the class did not enroll for either of the two hobby classes? 5% 15% 0% 25% Answer : b) wwwbiotecnika.org

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Q4. Of the 250 children who went to a park, 120 went by car, 90 took soft toys, and 150 had chocolates. 60 of them went by car and took soft toys, 20 had both chocolate and soft toys and 100 went by car and had chocolates and 30 had both soft toys and chocolate and went by car. How many children had neither chocolate nor soft toy and didn’t come by the car? 40 0 20 28 Answer: a) 40 wwwbiotecnika.org

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Q5. In a class of 100 students numbered 1 to 100, all the even numbered students opt for English, whose number are divisible by 5 opt for Hindi and those numbers are divisible by 7 opt for Kannada. How many opt for at least one of the three subjects? 56 66 67 57 Answer: b) 66 wwwbiotecnika.org

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Q6. Considering a house full show in a cinema hall with total 200 seats, 130 people ate salted popcorn, 150 people drank cold drink and 100 were seen eating popcorn along with cold drink. Find how many people were taking neither salted popcorn nor cold drink? 25 30 15 20 Answer : d) wwwbiotecnika.org

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Q7. There are 150 scientists in a lab. Out of which 65 had got exposed to UV rays, 40 got exposed to Benzene, another mutagen, and 30 to both the mutagens. Find the number of scientists exposed to (i) UV but not Benzene (ii) Benzene but not UV rays (iii) UV rays or Benzene (25, 15, 85) ( 40, 5, 100) (35, 10, 75) d) (5,10, 15 ) Answer: c) wwwbiotecnika.org

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Q8. If some cats are dogs and some dogs are wolfs. Then which statement might hold true? Only some cats are dogs. No cat is a wolf. Either a) or b) might be true. Both a) and b) might be true. Answer: d) wwwbiotecnika.org

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Q9. If All Z’s are X’s and all X’s are Y’s. Then which statement might hold true? Only some Y’s are Z’s. Only some X’s are Z’s. Either a) or b) might be true. Both a) and b) might be true. Answer: d) wwwbiotecnika.org

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Q 10. If All R’s are P’s and all P’s are Q’s. Then which statement might hold true? Only all R’s are Q’s. Only some R’s are not P’s. Either a) or b) might be true. Both a) and b) might be true. Answer: a) wwwbiotecnika.org

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Q11. If all N’s are L’s and all L’s are M’s. Then which statement might hold true? Only all L’s are N’s. Only all M’s are L’s. Either a) or b) might be true. Neither a) nor b) might be true. Answer: d) wwwbiotecnika.org

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Q12. If all tables are chairs and some chairs are bed. Then which statement might hold true? Only some beds are tables. Some chairs are tables. Either a) or b) might be true. Neither a) nor b) might be true. Both a) and b) might be true. Answer b) wwwbiotecnika.org

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Q13. If all bats are balls and some balls are wickets. Then which statement might hold true? Some wickets, if they are bats, they are also balls. All wickets which are not balls are also not bats. Neither a) or b) might be true. Both a) and b) might be true. Answer: d) wwwbiotecnika.org

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Q14. If no roses are sunflowers and some sunflowers are Daisies. Then which statement might hold true? Some Daisies are not sunflowers. Daisies which are not sunflowers are Roses. Neither a) or b) might be true. Both a) and b) might be true. Answer: a) wwwbiotecnika.org

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Q15. If no apples are oranges and some oranges are litchi. Then which statement might hold true? No litchi is an apple. Litchi which are oranges, are not apples. Either a) or b) might be true. Both a) and b) might be true. Answer: b) wwwbiotecnika.org

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Q16. If some crayons are red, and all red crayons are blue. Then which statement might hold true? Some crayons are blue. Some blue crayons are red. Either a) or b) might be true. Both a) and b) might be true. Answer: d) wwwbiotecnika.org

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Q17. If all the water is cold, and all milk is cold. Some water is milk. All cold is either milk or water. Neither a) or b) might be true. Both a) and b) might be true. Answer: a) wwwbiotecnika.org

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Q 18. Refer to the following figure to answer the following questions. In the following figure, the hexagon denotes doctors, the circle is for dancers, while the diamond consists of football players. A B C D G E F wwwbiotecnika.org

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i. Find the doctors who can dance and also play football in their past time. D E C A Answer a) wwwbiotecnika.org

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ii. Find the non-doctors who are both dancers and play football. D E C A Answer d) wwwbiotecnika.org

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Q19 i . Choose the diagram that best illustrates the relationship between shirts, shorts, sarees. B D A C Answer b) A C D B wwwbiotecnika.org

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Q19 ii . Choose the diagram that best illustrates the relationship between these three objects such as lakes, rain and water. A B C D Answer b) A C D B wwwbiotecnika.org

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Q20 : The question has a set of three statements (i, ii,iii ). Each statement has three segments. Chose the alternative where the third segment in the statement can be logically deduced from the preceding two, but not just one of them. All bats are mammals. No mammals are birds. No bats are birds. Some players are winner. All winners are humans. Some players are human. Some balls do not bounce. All ping pong bounce. Ping pong are not balls. i and ii Only i Only ii All of the above. Answer: a) wwwbiotecnika.org

Tips and tricks:

Tips and tricks Read the question very carefully and slowly and think about it. Keep in mind that any " givens”. Assign the objects numbers or alphabets. Some of the questions will be a variation from the commonly known facts. Every object in a set is unique. The same objects cannot be included in the set more than once. Use of word ‘between’ means that the range of numbers is not inclusive. In the statement of the problem, the word ‘or’ gives us a clue of union and the word ‘and’ gives us a clue of intersection. Frequently used formulae in set theory n (A U B) = n(A) + n(B) - n(A ∩ B ), n ( A ∪ B ∪ C ) = n ( A ) + n ( B ) + n ( C ) – n ( A ∩ B ) – n ( B ∩ C) – n ( A ∩ C ) + n ( A ∩ B ∩ C) n (A′ ∩ B′) = n (A ∪ B )′ = n (U) – n (A ∪ B) Keep in mind there can be multiple Venn diagrams for a single statement. wwwbiotecnika.org

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