Basic numeracy-set-theory-venn-diagrams-functions-relations

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Basic numeracy-set-theory-venn-diagrams-functions-relations

  1. 1. Click Here to Join Online Coaching Click Here www.upscportal.com Basic Numeracy
  2. 2. www.upscportal.com Click Here to Join Online Coaching Click Here Set Theory, Venn Diagrams Functions & Relations Sets A set is a collection of well defined objects. The objects of the sets are called elements. (i) Sets are usually denoted by capital letters A, B, C,..., X, Y, Z. (ii) The elements of the sets are denoted by small letters like a, b, c,..., x, y, z etc. Representation of Sets Sets are usually described into two ways. (i) Tabular form or roster form, in this form, all the elements of the set are separated by commas and enclosed between the bracket { }. For example (a) The set of vowels of English Alphabet as A = {a, e, i, o, u) (b) The set of numbers on a clock face is written as B = (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
  3. 3. www.upscportal.com Click Here to Join Online Coaching Click Here (ii) Set builder from: We define a set by stating properties which its elements must satisfy. For example the set of all even integers. Then, we use the letters usually x, and we write A = {x | x is an even integer} This is to be read as A is a set of numbers x such that x is an even integer. The vertical line “ | ” to be read as “such that” some times we use x in place of vertical line. A = {x : x is an even integer} eg, C = {1,w, w2} = {x | x3 – 1 = 0} If an object x is an element of a set A, we write x  A which is read as “x belong to A” and if an object x is not a member of A we write x  A and read as “x does not belong to A”. Some Important Terms (i) Empty or Null set The set which contains no elements is called the empty set or the null set. The empty set is written as f. Thus, f = { } as there is no element in the empty set. For example; the set of odd numbers divisible by 2 is the null set.
  4. 4. www.upscportal.com Click Here to Join Online Coaching Click Here (ii) Singleton set A set containing only one element is called a singleton for example, {1}, {4} are singleton sets. (iii) Equality of sets. The sets A and B are equal if they have same members that is if every elements of A is an element of B and every element of B is an element of A, then A = B eg, if A = { l, 3,5,7} and B = {7, 3, 1, 5}, then A = B If the two sets are not equal we write A  B Important Formulae 1. A set does not change if its elements, are repeated. 2. A set does not change even if the order of its elements is different. (iv) Finite and Infinite set. The set which contains a definite number of elements is called a finite set. The set which contains an infinite number of elements is called an infinite set. eg, (I) The set of days in a week. eg, (II) The set of natural numbers. (v) Disjoint set. Two sets A and B are said to be disjoint, if they do not have any element in common. eg, A = { 1, 2, 3}, B = { 4, 5, 6} are disjoint sets.
  5. 5. www.upscportal.com Click Here to Join Online Coaching Click Here (vi) Subset. If every element in set A is also an element of another set B. Then A is called a subset of B. Also B is said to be super set of A. Symbolically, we write A  B (ie, A contained in B) B  A (ie, B contains A) More specifically A  B if x A  x  B eg, (I) Let A = { 2, 4, 7}, B = { 1, 2, 3, 4, 7} Then, A Î B since every element of A is in B. eg, (II) A = {x | x a real number} and B = {x | x is an integer} Then, A  B 1. If there is at least one element of A which is not in B, then A is not a subset of B written as A  B. 2. Every set is a subset of itself ie, A  A. 3. If A  B and B c  A, then A = B. (vii) The Null set f is a subset of every set A. (viii) Proper Subset: A is a proper subset of B. if A  B and A  B and is written as A  B ie, if B contains at least one element more than A, then A is a proper subset of B
  6. 6. www.upscportal.com Click Here to Join Online Coaching Click Here (ix) Power set: Set of all the subsets of a set is called the power set eg, A = {a, b, c} subsets of A are f, {a}, {b}, {c}, {a, b}, {b, c}, {c,a}, {a, b, c} Hence, P(A) = [f, {a}, {b}, {c}, {a, b}, {b, c}, {c, a}, {a, b, c}] If n is the number of elements of a set A, then the number of subset of A ie, the number of elements of P (A) = 2n. (x) Universal set: If all the sets under consideration are the subsets of a fixed set U, then U is called the Universal set. Union of sets Union of two sets A and B is the set of all elements which belongs to A or B (or to both) and is written as A  B (ie, A union B) The same is defined in set builder form as A  B = {x|x  A or x  B} If A = {1, 3, 5, 7, 9} and B = {2, 4, 5, 6, 9} Then, A  B = { 1, 2, 3, 4, 5, 6, 7, 9}
  7. 7. www.upscportal.com Click Here to Join Online Coaching Click Here 1. From the definition of Union of sets A u B = B u A (Commutative Law) If A is any set, then A  A = A and A  f = A 2. If A and B are any two sets, then A  (A  B) and B  (A  B) If x Î A  B, then x  A or x  B and if x  A  B, then x  A and x  B. 3. If A, B, C are three sets, then A  (B  C) = (A  B)  C Intersection of Sets If A and B are any two sets, then intersection of A and B is the set of all elements which are in A and also in B. It is written as A Ç B and is read as “A intersection B‟ If A = {2, 4, 6, 8} and B = {4, 5, 6, 9} Then A  B = {4, 6} 1. From the definition of the intersection, it follows A  B = B  A (Commutative Law) 2. If A is any set, then A  A = A and A  f) = f 3. For any two sets A and B. A  B = A and A  B  B
  8. 8. www.upscportal.com Click Here to Join Online Coaching Click Here 4 If A and B have no elements in common ie, A and B are disjoint, then A  B = f If x  A  B = x  A and x  B eg, (I) If A = { 2, 3, 6, 8, 9} and B = (1, 3, 5, 6, 7, 9}, then A  B = {3, 6, 9} eg,(II) If A = {x1|< x < 4 } and B = {x|2 < x < 5}, then A  B = {x|2 < x < 4} If A, B, C are three sets, then (i) (A  B)  C = A  (B  C) Associative Law (ii) A  (B  C) = (A  B)  (A  C) Distributive Law Difference of Sets The difference of two sets A and B is set of elements which belongs to A but do not belong to B. This is written as A – B A – B = {x| x  A and x  B} 1.Set A – B subset of A ie, A – B  A 2. Set (A – B) and B are disjoint ie, (A – B)  B = f 3. A – B = (A  B) – (A  B)
  9. 9. www.upscportal.com Click Here to Join Online Coaching Click Here Symmetric Difference of Sets The symmetric difference of two sets A and B is (A – B)  (B – A) and is written as A D B Thus, A D B = (A – B)  (B – A) In the set builder form A D B = {x | x  A or x  B, but x  A  B} Demorgan Laws If A, B, C are three sets, then (i) A – (B  C) = (A – B)  (A – C) (ii) A – (B  C) = (A – B)  (A – C) Complement of a Set Let A be a subset of universal set U, then the complement of A is denoted by AC is defined by AC = {x  U, x  A} x  AC  x  A eg, (I) If U = {1, 2, 3, 4, 5, 6} and A = { 1, 3, 5}, then AC = {2, 4, 6} eg, (II) U be the set of all letters in English alphabet and A is a set of all vowels, then AC is the set of all consonants. 1. (A  B)C = AC  BC 2. (A  B)C = AC  BC
  10. 10. www.upscportal.com Click Here to Join Online Coaching Click Here Venn Diagrams A simple way of explaining the relation between sets is by a diagram which is called Venn diagram. In this a set is generally represented by a circle and its elements by points in the circle. Case I: A  U and B  U and A  B  f Here A and B are represented by a circle. A – B is the lined region B – A is dotted region and A  B is plane region.
  11. 11. www.upscportal.com Click Here to Join Online Coaching Click Here Case II: A = {a, c, e}, B = {b, d} A  B = f and A – B = A and B – A = B
  12. 12. www.upscportal.com Click Here to Join Online Coaching Click Here Case III: When A  B  U In adjoining figure, in Venn diagram A  B = B, A  B = A and A – B = f Some results from the Venn diagram (i) n(A  B) = n (a) + n(B) – n(A  B) (ii) n(A  B) = n (a) + n(B), when A  B = f (iii) n(A – B) + n (A  B) = n(A) (iv) n(B – A) + n (A  B) = n(A) (v) n(A – B) + n (A  B) + n (B – A) = n (A  B)
  13. 13. www.upscportal.com Click Here to Join Online Coaching Click Here Example 1: If in a factory of 30 workers, 10 take tea but not coffee and 14 take tea. Then how many take only coffee ? Solution. Total number of workers = n (T  C) = 30 Number of workers who take tea n (T) = 14 Who take tea but not coffee = n (T – C) = 10 Who drinks both coffee and tea = n (T) – n (T – C) = 14 – 10 = 4 Who takes only coffee = n (C – T) = x From the figure = x + 4 + 10 = 30 = x = 30 – 14 = 16 The worker who drinks only coffee = 16
  14. 14. www.upscportal.com Click Here to Join Online Coaching Click Here Example 2: An elocution competition was held in English and Hindi. Out of 80 students, 45 took part in English, 35 in Hindi, 15 in both English and Hindi, then for the number of students. (a) Who took part in English but not in Hindi. (b) Who took part in Hindi but not in English. (c) Who took part in either English or Hindi. (d) Who took part in neither.
  15. 15. www.upscportal.com Click Here to Join Online Coaching Click Here Solution. Suppose E is the set of students who took part in English, His the set of students who took part in Hindi, then E n H gives the set of students who took part in both English and Hindi. (a) The number of students who took part in English but not in Hindi = n(E) = n(E  H) = 45 – 15 = 30 (b) The number of students who took part in Hindi but not in English = n(H) – n(E  H) = 35 – 15 = 20 (c) The number of students who took part either in English or in Hindi is n(E  H) = n(E) + n(H) – n(E  H) = 45 + 35 – 15 = 65 (d) The number of students who took part neither in English nor in Hindi = n (S) –n (T  H) = 80 – 65 = 15 Ordered Pair If a, b be any two objects, then the pair (a, b) is called the ordered pair. The object a is called the first coordinate (or first number) and b is called the second coordinate (or second number) of the ordered pair (a, b). 1. The ordered pair (a, b)  (b, a) Two ordered pairs (a, b) and (c, d) are said to be equal, if and only if a = c and b = d.
  16. 16. www.upscportal.com Click Here to Join Online Coaching Click Here Cartesian Product of Sets If A and B be any two sets, then cartesian product of A and B is the set of all ordered pair (a, b), where a  A and b  B Cartesian product of A and B is written as A × B (ie, A cross B) ie A × B = {(a, b) | a Î A and b Î B} eg, If A = {a, b, c} and B = {1, 2}, then A × B = {(a, 1), (a, 2),(b, 1),(b, 2),(c, 1),(c, 2)} B × A = {(1, a),(1, b),(1, c),(2, a),(2, b),(2, c)} Thus, A × B ¹ B × A A × A = {(a, a), (a, b), (a, c),(b, a),(b, b),(b, c) (c, a),(c, b),(c, c)} B × B = {(1, 1), (1, 2),(2, 1),(2, 2)) 1. A × (B  C) = (A × B)  (A × C) 2. A ×(B  C) = (A × B)  (A × C) Relations A relation is a set of ordered pairs. If (x, y) is a member of a relation R, we write it as x R y (ie relation R to y). eg, If R is the ordered pairs of positive integers where R = {(x, y)| x2 = y} The relation is y is a square of x and the set is {(1, 1), (2, 4),(3, 9),(4, 16),...}
  17. 17. www.upscportal.com Click Here to Join Online Coaching Click Here Types of Relations (i) Reflexive: A relation R on a set A is said to be reflexive for every x  A (x, x)  R (ii) Symmetric Relation: A relation R on a set A is said to be symmetric if x R y  y R x (x, y)  R = (y,x)  R eg, Let A = {1, 2, 3} and R = {(1, 1), (2, 2),(1, 3),(3,1)} Clearly, R is a symmetric relation. (iii) Transitive Relation: A relation R in a set A is called transitive if x R y and y R z  x R z eg, Let R be a relation in the real number defined by “x less than y” then x < y and y < z = x < z (iv) Equivalence Relation A relation which is reflexive, symmetric and transitive is a equivalence relation.
  18. 18. www.upscportal.com Click Here to Join Online Coaching Click Here Functions If each element of a set A is associated with exactly one element in the set B, then this association is called a function from A to B. The set A is called the domain and the set B is called the co- domain of the function. Consider : A = {1, 2}, and B = (3, 4, 5, 6), then {(1, 4),(2, 5)) is a function {(1, 4),(2, 5),(2, 6)} is not a function since element 2 in the set A have two images 5 and 6 in the set B 1. Each element of A must be associated with exactly one element in the set B. 2. All the elements of the set B need not have the association. 3. The set of elements of B which are associated with the elements of the set A is called the „range‟ of the function. 4. The range is the subset of the co-domain. Types of Functions (i) One-one Function (injection): A function f : A  B is said to be a one-one function elements of A have different images in B ie,
  19. 19. www.upscportal.com Click Here to Join Online Coaching Click Here f(x) = z x, x  {1, 2, 3} f = {(1, 2),(2, 4),(3, 6)} (ii) Many-one Function: A function f : A  B is said to be a many-one function if two are n of A have the same images in B. (iii) Onto Function: A function f : A  B is called an onto function if every element of B is an image of some elements of A ie, if co- domain = range. eg, Let A = {a, b, c, d} and B = {1, 2, 3} f = {(a, 3),(b, 2),(c, 2),(d, 1)) (iv) Into Function: A function f: A – B is called an into function if co-domain  range. Example 3: A is set of prime numbers less than 20, write A in Roster form. Solution. Prime numbers Less than 20 are 2, 3, 5, 7, 11; 13, 17, 19 set A in Roster form. A = {2, 3, 5, 7, 11, 13, 17, 19}
  20. 20. www.upscportal.com Click Here to Join Online Coaching Click Here Example 4: Let A = {4, 5, 6, 7} and B = {6, 4, 7, 5}, then Solution. {4, 5, 6, 7} = {6, 4, 7, 5}, since each of the elements 4 , 5, 6, 7 belongs to Band each of the elements 6, 4, 7, 5 belongs to A, then A = B. The set does not change if its elements are rearranged. Example 5: A = {x2 = 16, x is odd}, then Solution. A is a empty set. x 2 = 16 Þ x = + 4 or x = –4, but x is not odd A does not contain any element, A = f Example 6: Rewrite the following statements using set notations. (a) x does not belongs to A (b) A is not a subset of B (c) H does not include D (d) d is a member of E. Solution. (a) x  A (b) A  B (c) H  D (d) d  E Example 7: Let A = {a, b, c}; ie, A contains the elements a, b, c, state whether each of the four statements is correct or incorrect tell why.
  21. 21. www.upscportal.com Click Here to Join Online Coaching Click Here (a) a Î A (b) a Í A (c) {a} Î A (d) {a} Í A Solution. (a) a  A, correct. (b) Incorrect. The symbol  must connect two set it indicates that one set a subset of other. Therefore, a  A is incorrect since a is a member of A, not a subset. (c) Incorrect. The symbol a connects an objects to a set. It indicates that object is a member of the set. Therefore, {a}  A is incorrect since {a} is a subset of A. (d) Correct. Example 8: If S be the universal set of English alphabet and let A = {a, b, c}, then complement of A is Solution. AC = {d, e, f … x, y, z} Example 9: If A = {1, 2, 3, 4} and B = {2, 4, 6, 8}, find A – B, B – A and A D B. Solution. A – B = {1, 3},
  22. 22. www.upscportal.com Click Here to Join Online Coaching Click Here (A – B) contains the elements of A but not the elements of B. Similarly, B – A = 16,81 (B – A) contains the elements of B but not the elements of A. A D B = (A – B)  (B – A) = {1, 3, 6, 8} Example 10: If S = {1, 2, 3, 4, 5}, A= {1, 2, 4}, B= {2, 4, 5} Find (a) A  B (b) A  B (c) BC (d) B – A (e) AC  B (f) A  BC (g) AC  BC (h) BC – AC (i) (A  B)C (j) (A  B)C Solution. (a) A Ç B = {1, 2, 4, 5} (b) A Ç B = {2, 4} (c) The complement of B consists of letters which are in S but not in B, therefore BC = {1, 3} (d) B – A consisted of elements in B which are not in A ie, B – A = {5} (e) AC = {3, 5} and B = {2, 4, 5}, therefore, AC  B = {2, 3, 4, 5} (f) A = {1, 2,4) and BC = {1, 3}, therefore, A  BC = {1, 2,3,4} (g) AC = {3, 5}, and BC = {1, 3} ; therefore, AC  BC = {3} (h) BC = {1, 3}, and AC = (3,5); therefore, BC – AC = {1} (i) A  B = {2, 4}, therefore, (A  B)C = {1, 3,5} (j) A  B = {1, 2, 4, 5}; therefore, (A  B)C = {3}
  23. 23. www.upscportal.com Click Here to Join Online Coaching Click Here Example 11: A = { 1, 2, 3} and B = {a, b}, then find A × B and B × A Solution. A × B = {{1, a}, {1, b}, {2, a}, {2, b), {3, a}, {3, b}} and B × A = {{a,1}, {a, 2}, {a, 3}, {b, l}, {b, 2}, {b, 3}} A × B  B × A since the ordered pair (1, a)  (a, 1) Example 12: If the set A contains 4 elements and set B contains 3 elements, then A × B contains Solution. The set A × B contains 12 elements.
  24. 24. www.upscportal.com Click Here to Join Online Coaching Click Here Click Here to Join Online Coaching: http://upscportal.com/civilservices/courses Click Here to Buy Study Material in Hard Copy: http://upscportal.com/civilservices/study-kit

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