Example 19 - Chapter 1 Class 12 Relation and Functions
Last updated at Dec. 16, 2024 by Teachoo
Last updated at Dec. 16, 2024 by Teachoo
Example 19 Let R be a relation on the set A of ordered pairs of positive integers defined by (x, y) R (u, v) if and only if xv = yu. Show that R is an equivalence relation. If (x, y) R (u, v) , then xv = yu Check Reflexive If (x, y) R (x, y), then xy = yx Since, xy = yx Hence , R is reflexive. Check symmetric If (x, y) R (u, v) , then xv = yu Now, If (u, v) R (x, y) , then uy = vx Since, xv = yu, vx = uy ∴ uy = vx So, if (x, y) R (u, v) , then (u, v) R (x, y) So, R is symmetric. If (x, y) R (u, v) , then xv = yu If (u, v) R (a, b) , then ub = va u = 𝑣𝑎/𝑏 We need to prove that (x, y) R (a, b) , i.e. xb = ya Check transitive Putting (2) in (1) xv = yu xv = y(𝑣𝑎/𝑏) xvb = yva xb = ya Hence (x, y) R (a, b) So, if (x, y) R (u, v) & (u, v) R (a, b) , then (x, y) R (a, b) Thus R is transitive. Thus, R is an equivalence relation.
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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo