Check sibling questions

Ex 1.2, 8 - Show that f: A x B -> B x a, f(a, b) = (b, a) is bijective

Ex 1.2 , 8 - Chapter 1 Class 12 Relation and Functions - Part 2
Ex 1.2 , 8 - Chapter 1 Class 12 Relation and Functions - Part 3
Ex 1.2 , 8 - Chapter 1 Class 12 Relation and Functions - Part 4

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Transcript

Ex 1.2, 8 (Introduction) Let A and B be sets. Show that f: A × B → B × A such that f(a, b) = (b, a) is bijective function. Taking example Let A = {1, 2}, B = {3, 4, 5} A × B = { (1, 3), (1, 4), (1, 5), (2, 3), (2, 4), (2, 5) } f(1, 3) = (3, 1) f(1, 4) = (4, 1) f(1, 5) = (5, 1) f(2, 3) = (3, 2) f(2, 4) = (4, 2) f(2, 5) = (4, 1) B × A = { (3, 1), (3, 2), (4, 1), (4, 2), (5, 1), (5, 2) } Ex 1.2, 8 Let A and B be sets. Show that f: A × B → B × A such that f (a, b) = (b, a) is bijective function. f(a, b) = (b, a). We can say that f(x) = (b, a). where x = (a, b) Checking one-one(injective) f (x1) = (b1, a1) f (x2) = (b2, a2) Rough One-one Steps: 1. Calculate f(x1) 2. Calculate f(x2) 3. Putting f(x1) = f(x2) we have to prove x1 = x2 Putting f (x1) = f (x2) ⇒ (b1, a1) = (b2, a2) Hence, b1 = b2 & a1 = a2 Now, since a1 = a2 & b1 = b2 We can say that, (a1, b1) = (a2, b2) Hence, if f(x1) = f(x2) , then x1 = x2 Hence, f is one-one Check onto f: A × B → B × A f(a, b) = (b, a) f(x) = (b, a) Let y = (b, a) Now, for every (b, a) ∈ B × A, there exists (a, b) ∈ A × B, such that f(x) = y This is possible for all a ∈ A, and b ∈ B ∴ f is onto. Hence, f is one-one and onto i.e. bijective.

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.