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Mathematics · Class 11 Science

Ch 14Sets and Relations — Class 11 Mathematics, concept-first.

A set is one of the most basic building blocks of mathematics, and this chapter formalises an idea you already use informally. The concept of a set as a rigorous mathematical object was developed by the German mathematician Georg Cantor (1845-1918).

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Sets and their Representations

A set is a collection of well-defined objects — a collection where it is always possible to say for certain whether any given object belongs to it or not (this rules out vague groupings like 'clever students', where diff…

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5.1

Introduction

A set is one of the most basic building blocks of mathematics, and this chapter formalises an idea you already use informally.

+Exercise 5.1i55 questions
  1. Q1Describe the following set in Roster form: A = {x/x is a letter of the word 'MOVEMENT'}.Free
  2. Q2Describe the following set in Roster form: B = {x/x is an integer, $-\dfrac{3}{2} < x < \dfrac{9}{2}$}.Free
  3. Q3Describe the following set in Roster form: C = {x/x = 2n + 1, n∈N}.Free
  4. Q4Describe the following set in Set-Builder form: {0}.Preview
  5. Q5Describe the following set in Set-Builder form: {0, ±1, ±2, ±3}.Preview
  6. Q6Describe the following set in Set-Builder form: $\left\{\dfrac{1}{2}, \dfrac{2}{5}, \dfrac{3}{10}, \dfrac{4}{17}, \dfrac{5}{26}, \dfrac{6}{3…Preview
  7. Q7Describe the following set in Set-Builder form: {0, -1, 2, -3, 4, -5, 6, ...}.Preview
  8. Q8If A = {x/6x²+x-15 = 0}, B = {x/2x²-5x-3 = 0}, C = {x/2x²-x-3 = 0} then find (A∪B∪C).Preview
  9. Q9If A = {x/6x²+x-15 = 0}, B = {x/2x²-5x-3 = 0}, C = {x/2x²-x-3 = 0} then find (A∩B∩C).Preview
  10. Q10If A, B, C are the sets for the letters in the words 'college', 'marriage' and 'luggage' respectively, then verify that [A-(B∪C)] = [(A-B)∩(…Preview
  11. Q11If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: A∪(B∩C) = (A∪…Preview
  12. Q12If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: A∩(B∪C) = (A∩…Preview
  13. Q13If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: (A∪B)' = (A'∩…Preview
  14. Q14If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: (A∩B)' = A'∪B…Preview
  15. Q15If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: A = (A∩B)∪(A∩…Preview
  16. Q16If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: B = (A∩B)∪(A'…Preview
  17. Q17If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: (A∪B) = (A−B)…Preview
  18. Q18If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: A ∩ (BΔC) = (…Preview
  19. Q19If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: n(A∪B) = n(A)…Preview
  20. Q20If A = {1, 2, 3, 4}, B = {3, 4, 5, 6}, C = {4, 5, 6, 7, 8} and universal set X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, then verify: n(B) = n(A'∩B…Preview
  21. Q21If A and B are subsets of the universal set X and n(X) = 50, n(A) = 35, n(B) = 20, n(A'∩B') = 5, find n(A∪B).Preview
  22. Q22If A and B are subsets of the universal set X and n(X) = 50, n(A) = 35, n(B) = 20, n(A'∩B') = 5, find n(A∩B).Preview
  23. Q23If A and B are subsets of the universal set X and n(X) = 50, n(A) = 35, n(B) = 20, n(A'∩B') = 5, find n(A'∩B).Preview
  24. Q24If A and B are subsets of the universal set X and n(X) = 50, n(A) = 35, n(B) = 20, n(A'∩B') = 5, find n(A∩B').Preview
  25. Q25In a class of 200 students who appeared certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET…Preview
  26. Q26In a class of 200 students who appeared certain examinations, 35 students failed in CET, 40 in NEET and 40 in JEE, 20 failed in CET and NEET…Preview
  27. Q27From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and E…Preview
  28. Q28From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and E…Preview
  29. Q29From amongst 2000 literate individuals of a town, 70% read Marathi newspapers, 50% read English newspapers and 32.5% read both Marathi and E…Preview
  30. Q30In a hostel, 25 students take tea, 20 students take coffee, 15 students take milk, 10 student take both tea and coffee, 8 students take both…Preview
  31. Q31There are 260 persons with a skin disorder. If 150 had been exposed to the chemical A, 74 to the chemical B, and 36 to both chemicals A and…Preview
  32. Q32There are 260 persons with a skin disorder. If 150 had been exposed to the chemical A, 74 to the chemical B, and 36 to both chemicals A and…Preview
  33. Q33There are 260 persons with a skin disorder. If 150 had been exposed to the chemical A, 74 to the chemical B, and 36 to both chemicals A and…Preview
  34. Q34Write down the power set of A = {1,2,3}.Preview
  35. Q35Write the following interval in Set-Builder form: (-3, 0).Preview
  36. Q36Write the following interval in Set-Builder form: [6, 12].Preview
  37. Q37Write the following interval in Set-Builder form: (6, ∞).Preview
  38. Q38Write the following interval in Set-Builder form: (-∞, 5].Preview
  39. Q39Write the following interval in Set-Builder form: (2, 5].Preview
  40. Q40Write the following interval in Set-Builder form: [-3, 4).Preview
  41. Q41A college awarded 38 medals in volley ball, 15 in football and 20 in basket ball. The medals awarded to a total of 58 players and only 3 pla…Preview
  42. Q42Solve the following inequality and write the solution set using interval notation: −9 < 2x + 7 ≤ 19.Preview
  43. Q43Solve the following inequality and write the solution set using interval notation: x² − x > 20.Preview
  44. Q44Solve the following inequality and write the solution set using interval notation: $\dfrac{2x}{x-4} \le 5$.Preview
  45. Q45Solve the following inequality and write the solution set using interval notation: 6x² + 1 ≤ 5x.Preview
  46. Q46If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A∪B.Preview
  47. Q47If A = (-7, 3], B = [2, 6] and C = [4, 9] then find B∪C.Preview
  48. Q48If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A∪C.Preview
  49. Q49If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A∩B.Preview
  50. Q50If A = (-7, 3], B = [2, 6] and C = [4, 9] then find B∩C.Preview
  51. Q51If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A∩C.Preview
  52. Q52If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A'∩B.Preview
  53. Q53If A = (-7, 3], B = [2, 6] and C = [4, 9] then find B'∩C'.Preview
  54. Q54If A = (-7, 3], B = [2, 6] and C = [4, 9] then find B-C.Preview
  55. Q55If A = (-7, 3], B = [2, 6] and C = [4, 9] then find A-B.Preview
5.1.1

Set: Definition

A collection of well-defined objects is called a SET. Each object inside a set is called its ELEMENT or MEMBER.

5.1.2

Representation of a Set

There are three standard ways to describe or represent a set:

5.1.3

Number of Elements of a Set

The number of DISTINCT elements contained in a finite set A is denoted by , and is also called the CARDINALITY (or cardinal number) of the set A.

5.1.4

Types of Sets

This section catalogues the standard vocabulary used to classify sets.

5.1.5.1

Complement of a Set

The COMPLEMENT of a set A, denoted or , is defined as — that is, the set of all elements of the universal set U that are NOT in A.

5.1.5.2

Union of Sets

The UNION of two sets A and B is the set of all elements that are in A OR in B (using 'or' in the inclusive sense — an element in both A and B still counts), denoted . Formally, .

5.1.5.3

Intersection of Sets

The INTERSECTION of two sets A and B is the set of all elements that are in BOTH A and B, denoted . Formally, .

5.1.5.4

Difference and Symmetric Difference of Sets

The DIFFERENCE of set A and set B is the set of elements that are in A but NOT in B, denoted (also written ). Formally, (shaded in Fig. 5.6). Similarly, . For example, if and , then and .

5.1.5.5

Cardinality Properties of Sets

Given sets A, B (and sometimes a third set C, all subsets of a universal set), the following cardinality (counting) formulas hold: 1) — the general two-set union formula.

5.1.5.6

Maximum and Minimum of n(A∪B) and n(A∩B)

When only and are known (but not the exact overlap between A and B), the union and intersection counts are still bounded within a predictable range: (a) — the union can never be smaller than the large…

5.1.6

Intervals and Solving Inequalities

An interval is a special kind of subset of the real numbers R, consisting of all real numbers lying between two fixed bounds. Six standard types:

5.2

Relations

Having built up the vocabulary of sets, the chapter now turns to RELATIONS — a way of formally capturing how elements of one set connect to elements of another (or the same) set.

+Exercise 5.2i25 questions
  1. Q56If (x−1, y+4) = (1, 2) find the values of x and y.Free
  2. Q57If $\left(x + \dfrac{1}{3}, \dfrac{y}{3} - 1\right) = \left(\dfrac{1}{2}, \dfrac{3}{2}\right)$, find x and y.Free
  3. Q58If A = {a, b, c}, B = {x, y} find A×B, B×A, A×A, B×B.Free
  4. Q59If P = {1, 2, 3} and Q = {1,4}, find sets P × Q and Q × P.Preview
  5. Q60Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Verify A × (B ∩ C) = (A × B) ∩ (A × C).Preview
  6. Q61Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Verify A × (B ∪ C) = (A × B) ∪ (A × C).Preview
  7. Q62Express {(x, y) / x² + y² = 100, where x, y ∈ W} as a set of ordered pairs.Preview
  8. Q63Let A = {6, 8} and B = {1, 3, 5}. Show that R1 = {(a, b)/a∈ A, b∈B, a−b is an even number} is a null relation.Preview
  9. Q64Let A = {6, 8} and B = {1, 3, 5}. Show that R2 = {(a, b)/a∈ A, b∈B, a+b is odd number} is an universal relation.Preview
  10. Q65Write the relation in the Roster form. State its domain and range. R1 = {(a, a²) / a is prime number less than 15}.Preview
  11. Q66Write the relation in the Roster form. State its domain and range. R2 = {(a, 1/a) / 0<a≤5, a∈N}.Preview
  12. Q67Write the relation in the Roster form. State its domain and range. R3 = {(x, y) / y = 3x, y∈ {3, 6, 9, 12}, x∈ {1, 2, 3}}.Preview
  13. Q68Write the relation in the Roster form. State its domain and range. R4 = {(x, y) / y > x+1, x = 1, 2 and y = 2, 4, 6}.Preview
  14. Q69Write the relation in the Roster form. State its domain and range. R5 = {(x, y) / x+y = 3, x, y∈ {0, 1, 2, 3}}.Preview
  15. Q70Write the relation in the Roster form. State its domain and range. R6 = {(a, b) / a ∈N, a < 6 and b = 4}.Preview
  16. Q71Write the relation in the Roster form. State its domain and range. R7 = {(a, b) / a, b ∈N, a + b = 6}.Preview
  17. Q72Write the relation in the Roster form. State its domain and range. R8 = {(a, b) / b = a + 2, a ∈z, 0 < a < 5}.Preview
  18. Q73[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ Z, a−b is an integer}.Preview
  19. Q74[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ N, a+b is even}.Preview
  20. Q75[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ N, a divides b}.Preview
  21. Q76[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ N, a² − 4ab + 3b² = 0}.Preview
  22. Q77[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a is sister of b and a,b ∈ G = Set…Preview
  23. Q78[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : Line a is perpendicular to line b i…Preview
  24. Q79[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ R, a < b}.Preview
  25. Q80[Activity] Identify whether the following relation is reflexive, symmetric, and transitive: R = {(a,b) : a,b ∈ R, a ≤ b³}.Preview
5.2.1

Ordered Pair

A pair of numbers, where the ORDER in which the numbers appear matters, is called an ORDERED PAIR. The ordered pairs and are generally different (unless ).

5.2.2

Cartesian Product of Two Sets

Let A and B be two non-empty sets. The CARTESIAN PRODUCT of A and B, denoted , is defined as the set of ALL ordered pairs such that and : .

5.2.3

Cartesian Product of a Set with Itself

When a set is multiplied by itself, , where is an ordered pair. Extending further, , where is an ORDERED TRIPLET. For example, if , then (4 pairs), and (8 triplets).

5.2.4

Relation, Domain, Co-domain and Range

The everyday word 'relation' — two objects or quantities are 'related' if there is a recognisable link between them (A is a friend of B, F is the father of S, P is the sister of Q) — is formalised in…

5.2.5

Binary Relation on a Set

Let A be a non-empty set. Every SUBSET of is called a BINARY RELATION on A.

5.2.6

Identity Relation

If a relation on A is given by for EVERY — that is, the relation consists exactly of the pairs (sometimes called the DIAGONAL subset of ) — this is called the IDENTITY RELATION on A.

5.2.7

Types of Relations

Let A be a non-empty set. A binary relation R on A is said to be:

More questions

37 Q
+Show 10 questions10 questions
  1. Q81For the set A = {a, b, c, d, e} the correct statement is: A) {a, b} ∈ A B) {a}∈ A C) a∈ A D) a∉ AFree
  2. Q82If aN = {ax : x ∈N}, then set 6N∩8N = A) 8N B) 48N C) 12N D) 24NFree
  3. Q83If set A is empty set then n[P [P [P (A)]]] is A) 6 B) 16 C) 2 D) 4Free
  4. Q84In a city 20% of the population travels by car, 50% travels by bus and 10% travels by both car and bus. Then, persons travelling by car or b…Preview
  5. Q85If the two sets A and B are having 43 elements in common, then the number of elements common to each of the sets A × B and B × A is: A) 43²…Preview
  6. Q86Let R be a relation on the set N be defined by {(x, y) / x, y∈ N, 2x + y = 41} Then R is A) Reflexive B) Symmetric C) Transitive D) None of…Preview
  7. Q87The relation \">\" in the set of N (Natural number) is A) Symmetric B) Reflexive C) Transitive D) Equivalent relationPreview
  8. Q88A relation between A and B is A) only A × B B) An Universal set of A × B C) An equivalent set of A × B D) A subset of A × BPreview
  9. Q89If (x,y) ∈ N × N, then xy = x² is a relation which is A) Symmetric B) Reflexive C) Transitive D) EquivalencePreview
  10. Q90If A = {a, b, c} The total no. of distinct relations in A × A is A) 3 B) 9 C) 8 D) 2^9Preview
+Show 27 questions27 questions
  1. Q91Write down the following set in set builder form: {10, 20, 30, 40, 50}.Free
  2. Q92Write down the following set in set builder form: {a, e, i, o, u}.Free
  3. Q93Write down the following set in set builder form: {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}.Free
  4. Q94If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8, 9, 12}. Write down the set A∪B.Preview
  5. Q95If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8 9, 12}. Write down the set B∩C.Preview
  6. Q96If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8 9, 12}. Write down the set A-B.Preview
  7. Q97If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8 9, 12}. Write down the set B∩C'.Preview
  8. Q98If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8 9, 12}. Write down the set A∪B∪C.Preview
  9. Q99If U = {x/x∈N, 1 ≤ x ≤ 12}, A = {1, 4, 7, 10}, B = {2, 4, 6, 7, 11}, C = {3, 5, 8 9, 12}. Write down the set A∩(B∪C).Preview
  10. Q100In a survey of 425 students in a school, it was found that 115 drink apple juice, 160 drink orange juice and 80 drink both apple as well as…Preview
  11. Q101In a school there are 20 teachers who teach Mathematics or Physics. Of these, 12 teach Mathematics and 4 teach both Physics and Mathematics.…Preview
  12. Q102If A = {1, 2, 3} and B = {2, 4}, state the elements of A × A, A × B, B × A, B × B, (A × B) ∩ (B × A).Preview
  13. Q103If A = {−1, 1}, find A × A × A.Preview
  14. Q104If A = {1, 2, 3}, B = {4, 5, 6} check if the following is a relation from A to B. Also write its domain and range: R1 = {(1, 4), (1, 5), (1,…Preview
  15. Q105If A = {1, 2, 3}, B = {4, 5, 6} check if the following is a relation from A to B. Also write its domain and range: R2 = {(1, 5), (2, 4), (3,…Preview
  16. Q106If A = {1, 2, 3}, B = {4, 5, 6} check if the following is a relation from A to B. Also write its domain and range: R3 = {(1, 4 ), (1, 5), (3…Preview
  17. Q107If A = {1, 2, 3}, B = {4, 5, 6} check if the following is a relation from A to B. Also write its domain and range: R4 = {(4, 2), (2, 6), (5,…Preview
  18. Q108Determine the domain and range of the following relation: R = {(a, b) / a ∈ N, a < 5, b = 4}.Preview
  19. Q109Determine the domain and range of the following relation: R = {(a, b) / b = |a−1|, a ∈ Z, |a| < 3}.Preview
  20. Q110Find R : A→A when A = {1,2,3,4} such that R = {(a, b) / a−b = 10}.Preview
  21. Q111Find R : A→A when A = {1,2,3,4} such that R = {(a, b) / |a − b| ≥ 0}.Preview
  22. Q112R = {1,2,3} → {1,2,3} given by R = {(1,1) (2,2), (3,3), (1,2) (2,3)} Check if R is a) reflexive b) symmetric c) transitive.Preview
  23. Q113Check if R : Z → Z, R = {(a, b) / 2 divides a-b} is equivalence relation.Preview
  24. Q114Show that the relation R in the set A = {1, 2, 3, 4, 5} Given by R = {(a, b) / a−b is even} is an equivalence relation.Preview
  25. Q115Show that the following is an equivalence relation: R in A is the set of all books, given by R = {(x, y) / x and y have same number of pages…Preview
  26. Q116Show that the following is an equivalence relation: R in A = {x∈Z | 0 ≤ x ≤ 12} given by R = {(a, b) / |a−b| is a multiple of 4}.Preview
  27. Q117Show that the following is an equivalence relation: R in A = {x∈ N/x ≤10} given by R = {(a, b) / a=b}.Preview