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Last updated at Jan. 28, 2020 by Teachoo

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Misc 14 Define a binary operation *on the set {0, 1, 2, 3, 4, 5} as a * b = {โ(๐+๐, ๐๐ ๐+๐<6@&๐+๐ โ6, ๐๐ ๐+๐โฅ6)โค Show that zero is the identity for this operation and each element a โ 0 of the set is invertible with 6 โ 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 โฅ 6 Putting b = 0 a โฅ 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 โ a being the inverse of a. a * b = {โ(๐+๐, ๐๐ ๐+๐<6@&๐+๐ โ6, ๐๐ ๐+๐โฅ6)โค 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 โฅ 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 โ a being the inverse of a. s

Binary operations: Inverse

Chapter 1 Class 12 Relation and Functions

Concept wise

- Relations - Definition
- Empty and Universal Relation
- To prove relation reflexive, transitive, symmetric and equivalent
- Finding number of relations
- Function - Definition
- To prove one-one & onto (injective, surjective, bijective)
- Composite functions
- Composite functions and one-one onto
- Finding Inverse
- Inverse of function: Proof questions
- Binary Operations - Definition
- Whether binary commutative/associative or not
- Binary operations: Identity element
- Binary operations: Inverse

About the Author

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.