Miscellaneous

Chapter 1 Class 12 Relation and Functions
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### Transcript

Misc 12 Consider the binary operations * : R × R → and o : R × R → R defined as a * b = a – b﷯ and a o b= a, ∀ a, b ∈ R. Show that * is commutative but not associative, o is associative but not commutative. Further, show that ∀ a, b, c ∈ R, a * (b o c) = (a * b) o (a * c). (If it is so, we say that the operation * distributes over the operation o). Does o distribute over *? Justify your answer. Check commutative for * * is commutative if a * b = b * a Since a * b = b * a ∀ a, b ∈ R * is commutative Check associative for * * is associative if (a * b) * c = a * (b * c) Since (a * b) * c ≠ a * (b * c) * is not associative a o b = a Check commutative for o o is commutative if a o b = b o a Since a o b ≠ b o a * is not commutative Check associative for o o is associative if (a o b) o c = a o (b o c) Since (a o b) o c = a o (b o c) o is not associative a * b = a – b﷯ & a o b = a * distributes over o If a * (b o c) = (a * b) o (a * c), ∀ a, b, c ∈ R * distributes over o Since a * (b o c) = (a * b) o (a * c), ∀ a, b, c ∈ R * distributes over o a * b = a – b﷯ & a o b = a o distributes over * If a o (b * c) = (a o b) * (a o c), ∀ a, b, c ∈ R o distributes over * Since a o (b * c) ≠ (a o b) * (a o c) o does not distributes over *

#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.