The notion of sets provides the stimulus for learning higher concepts in mathematics. A set is a collection of well-defined objects. This means that a set is merely a collection of something which we may recognize. In this chapter, we try to extend the concept of sets in two forms called Relations and Functions. For doing this, we need to first know about cartesian products that can be defined between two non-empty sets.
Question-1: Find A × B, A × A and B × A (i) A = {2, -2, 3} and B = {1, -4} (ii) A = B = {p, q} (iii) A – {m, n} ; B = Φ
Solution:
(i) A = {2, -2, 3} and B = {1, -4} A × B = {2,-2, 3} × {1,-4} = {(2, 1), (2, -4)(-2, 1) (-2, -4) (3, 1) (3,-4)} A × A = {2,-2, 3} × {2,-2, 3} = {(2, 2)(2, -2)(2, 3)(-2, 2) (-2, -2)(-2, 3X3,2) (3,-2) (3,3)} B × A = {1,-4} × {2,-2, 3} = {(1, 2)(1, -2)( 1, 3)(-4, 2) (-4,-2)(-4, 3)}
ii) A = B = {p, q} A × B = {p, q) × {p, q} = {(p,p),(p,q)(q,p)(q,q)} A × A = {p,q) × (p,q) = {(p,p)(p,q)(q,p)(q,q) B × A = {p,q} × {p,q} = {(p,p)(p,q)(q,p)(q,q)
(iii) A = {m, n} × B = Φ Note: B = Φ or {} A × B = {m, n) × { } = { ) A × A = {m, n) × (m, n)} = {(m, m)(w, w)(n, m)(n, n)} B × A = { } × {w, n} = { }
Question-2: Let A = {1, 2, 3} and B = {x | x is a prime number less than 10}. Find A × B and B × A.
Question-3: If B × A = {(-2, 3),(-2, 4),(0, 3),(0, 4), (3,3) ,(3, 4)} find A and B.
Solution: B × A = {(-2, 3)(-2, 4) (0, 3) (0, 4) (3, 3) (3,4)} A = {3,4} B = {-2,0,3}
Question-4: If A ={5, 6}, B = {4, 5, 6} , C = {5, 6, 7}, Show that A × A = (B × B) ∩ (C × C).
Solution: A = {5,6}, B = {4, 5, 6},C = {5, 6, 7} A × A = {(5, 5), (5, 6), (6, 5), (6, 6)} ……….. (1) B × B = {(4, 4), (4, 5), (4, 6), (5, 4), (5, 5), (5, 6), (6, 4), (6, 5), (6, 6)} …(2) C × C = {(5, 5), (5, 6), (5, 7), (6, 5), (6, 6), (6, 7), (7, 5), (7, 6), (7, 7)} …(3) (B × B) ∩ (C × C) = {(5, 5), (5,6), (6, 5), (6,6)} …(4) (1) = (4) A × A = (B × B) ∩ (C × C) It is proved.
Question-5: Given A = {1,2,3}, B = {2,3,5}, C = {3,4} and D = {1,3,5}, check if (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) is true?
Solution: A = {1,2, 3}, B = {2, 3, 5}, C = {3,4} D = {1,3,5} A ∩ c = {1,2,3} ∩ {3,4} = (3} B ∩ D = {2,3, 5} ∩ {1,3,5} = {3,5} (A ∩ C) × (B ∩ D) = {3} × {3,5} = {(3, 3)(3, 5)} ….(1) A × B = {1,2,3} × {2,3,5} = {(1,2) (1,3) (1,5) (2, 2) (2, 3) (2, 5) (3, 2) (3, 3) (3, 5)} C × D = {3,4} × {1,3,5} = {(3,1) (3, 3) (3, 5) (4,1) (4, 3) (4, 5)} (A × B) ∩ (C × D) = {(3, 3) (3, 5)} ….(2) From (1) and (2) we get (A ∩ C) × (B ∩ D) = (A × B) ∩ (C × D) This is true.
Question-6: Let A = {x ∈ W | x < 2}, B = {x ∈ N |1 < x < 4} and C = {3, 5}. Verify that (i) A × (B ∪ C) = (A × B) ∪ (A × C) (ii) A × (B ∩ C) = (A × B) ∩ (A × C) (iii) (A ∪ B) × C = (A × C) ∪ (B × C) (iv) A × (B ∪ C) = (A × B) ∪ (A × C)
Solution: (i) A = {x ∈ W|x < 2} = {0,1} B = {x ∈ N |1 < x < 4} = {2,3,4} C = {3,5} LHS =A × (B ∪ C) B ∪ C = {2, 3, 4} ∪ {3, 5} = {2, 3, 4, 5} A × (B ∪ C) = {(0, 2), (0, 3), (0,4), (0, 5), (1, 2) , (1, 3), (1, 4),(1, 5)} ………. (1) RHS = (A × B) ∪ (A × C) (A × B) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} (A × C) = {(0, 3), (0, 5), (1, 3), (1, 5)} (A × B) ∪ (A × C)= {(0, 2), (0, 3), (0,4), (1, 2), (1, 3), (1, 4), (0, 5), (1, 5)} ….(2) (1) = (2), LHS = RHS Hence it is proved.
(ii) A × (B ∩ C) = (A × B) ∩ (A × C) LHS = A × (B ∩ C) (B ∩ C) = {3} A × (B ∩ C) = {(0, 3), (1, 3)} …(1) RHS = (A × B) ∩ (A × C) (A × B) = {(0, 2), (0, 3), (0, 4), (1, 2), (1, 3), (1, 4)} (A × C) = {(0, 3), (0, 5), (1, 3), (1, 5)} (A × B) ∩ (A × C) = {(0, 3), (1, 3)} ……….. (2) (1) = (2) ⇒ LHS = RHS. Hence it is verified.
(iii) (A ∪ B) × C = (A × C) ∪ (B × C) LHS = (A ∪ B) × C A ∪ B = {0, 1, 2, 3, 4} (A ∪ B) × C = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} …………. (1) RHS = (A × C) ∪ (B × C) (A × C) = {(0, 3), (0, 5), (1, 3), (1, 5)} (B × C) = {(2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} (A × C) ∪ (B × C) = {(0, 3), (0, 5), (1, 3), (1, 5), (2, 3), (2, 5), (3, 3), (3, 5), (4, 3), (4, 5)} ………… (2) (1) = (2) ∴ LHS = RHS. Hence it is verified.
Question-7: Let A = The set of all natural numbers less than 8, B = The set of all prime numbers less than 8, C = The set of even prime number. Verify that (i) (A ∩ B) × C = (A × c) ∩ (B × C) (ii) A × (B – C) = (A × B) – (A × C)
Solution: A = {1,2, 3, 4, 5,6, 7} B = {2, 3, 5,7} C = {2}
(i) (A ∩ B) × C = (A × C) ∩ (B × C) A ∩ B = {1, 2, 3, 4, 5, 6, 7} ∩ {2, 3, 5, 7} = {2, 3, 5, 7} (A ∩ B) × C = {2, 3, 5, 7} × {2} = {(2, 2) (3, 2) (5, 2) (7, 2)} ….(1) A × C = {1,2, 3, 4, 5, 6, 7} × {2} = {(1,2) (2, 2) (3, 2) (4, 2) (5.2) (6, 2) (7, 2)} B × C = {2, 3, 5, 7} × {2} = {(2, 2) (3, 2) (5, 2) (7, 2)} (A × C) ∩ (B × C) = {(2, 2) (3, 2) (5, 2) (7, 2)} ….(2) From (1) and (2) we get (A ∩ B) × C = (A × C) ∩ (B × C)
(ii) A × (B – C) = (A × B) – (A × C) B – C = {2, 3, 5, 7} – {2} = {3,5,7} A × (B – C) = {1,2, 3, 4, 5, 6,7} × {3,5,7} = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7) (7, 3) (7, 5) (7, 7)} ………….(1) A × B = {1,2, 3, 4, 5, 6, 7} × {2, 3, 5,7} = {(1, 2) (1, 3) (1, 5) (1, 7) (2, 2) (2, 3) (2, 5) (2, 7) (3, 2) (3, 3) (3, 5) (3, 7) (4, 2) (4, 3) (4, 5) (4, 7) (5, 2) (5, 3) (5, 5) (5, 7) (6, 2) (6, 3) (6, 5) (6, 7) (7, 2) (7, 3) (7, 5) (7, 7)} A × C = {1,2, 3,4, 5, 6, 7} × {2} = {(1, 2) (2, 2) (3, 2) (4, 2) (5, 2) (6.2) (7,2)} (A × B) – (A × C) = {(1, 3) (1, 5) (1, 7) (2, 3) (2, 5) (2, 7) (3, 3) (3, 5) (3, 7) (4, 3) (4, 5) (4, 7) (5, 3) (5, 5) (5, 7) (6, 3) (6, 5) (6, 7) (7, 3) (7, 5) (7, 7)} ….(2) From (1) and (2) we get A × (B – C) = (A × B) – (A × C)
Relations: Let A and B be any two non-empty sets. A “relation” R from A to B is a subset of A × B satisfying some specified conditions.
Note: The domain of the relations R = {x ∈ A/xRy, for some y ∈ B} The co-domain of the relation R is B The range of the ralation R = (y ∈ B/xRy for some x ∈ A}
