Question 22 Show that – a is the inverse of a for the addition operation ‘+’ on R and 1﷮a﷯ is the inverse of a ≠ 0 for the multiplication operation ‘×’ on R. Addition An element a in R is invertible if, there is an element b in R such that , a * b = e = b * a Here, b is the inverse of a Here, e = 0 for addition So, a * b = e = b * a a + b = 0 = b + a ⇒ b = –a Since a + (– a) = 0 = (– a) + a = 0, So, –a is the inverse of a for addition. Multiplication An element a in R is invertible if, there is an element b in R such that , a * b = e = b * a Here, b is the inverse of a Here, e = 1 for multiplication So, a * b = e = b * a a × b = 1 = b × a ⇒ b = 1﷮𝑎﷯ Since a × 1﷮𝑎﷯ = 1 = 1﷮𝑎﷯ × a So, 𝟏﷮𝒂﷯ is the inverse of a for addition.

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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.

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