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  1. Chapter 1 Class 12 Relation and Functions
  2. Serial order wise

Transcript

Misc 14 Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as a * b = {โ–ˆ(๐‘Ž+๐‘, ๐‘–๐‘“ ๐‘Ž+๐‘<6@&๐‘Ž+๐‘ โˆ’6, ๐‘–๐‘“ ๐‘Ž+๐‘โ‰ฅ6)โ”ค Show that zero is the identity for this operation and each element a โ‰  0 of the set is invertible with 6 โˆ’ a being the inverse of a. e is the identity of * if a * e = e * a = a Checking if zero is identity for this operation If a + b < 6 Putting b = 0 a < 6 This is possible Now, a * 0 = a + 0 = a 0 * a = 0 + a = a Thus, a * 0 = 0 * a = a So, 0 is identity of * If a + b โ‰ฅ 6 Putting b = 0 a โ‰ฅ 6 This is not possible as value of a can be {0, 1, ,2, 3, 4, 5} Now, we need to show that each element a โ‰  0 of the set is invertible with 6 โˆ’ a being the inverse of a. a * b = {โ–ˆ(๐‘Ž+๐‘, ๐‘–๐‘“ ๐‘Ž+๐‘<6@&๐‘Ž+๐‘ โˆ’6, ๐‘–๐‘“ ๐‘Ž+๐‘โ‰ฅ6)โ”ค An element a in set is invertible if, there is an element in set such that , a * b = e = b * a Putting b = 6 โ€“ a So, a + b = a + (6 โ€“ a) = 6 Since a + b โ‰ฅ 6 a * b = a + b โ€“ 6 a * b = a * (6 โ€“ a) = a + (6 โ€“ a) โ€“ 6 = 0 b * a = (6 โ€“ a) * a = (6 โ€“ a) + a โ€“ 6 = 0 Since a * (6 โ€“ a) = (6 โ€“ a) * a = 0 Hence, each element a of the set is invertible with 6 โˆ’ a being the inverse of a. s

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.