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 = ๐+๐, ๐๐ ๐+๐<6๏ทฎ&๐+๐ โ6, ๐๐ ๐+๐โฅ6๏ทฏ๏ทฏ 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 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 = ๐+๐, ๐๐ ๐+๐<6๏ทฎ&๐+๐ โ6, ๐๐ ๐+๐โฅ6๏ทฏ๏ทฏ 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 โ a being the inverse of a.