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Example 38 - Show that zero is identity for addition on R - Examples

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  1. Chapter 1 Class 12 Relation and Functions
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Example 38 Show that zero is the identity for addition on R and 1 is the identity for multiplication on R. But there is no identity element for the operations – : R × R → R and ÷ : R∗ × R∗ → R∗. Addition e is the identity of * if a * e = e * a = a i.e. a + e = e + a = a This is possible if e = 0 Since a + 0 = 0 + a = a ∀ a ∈ R 0 is the identity element for addition on R Multiplication e is the identity of * if a * e = e * a = a i.e. a × e = e × a = a This is possible if e = 1 Since a × 1 = 1 × a = a ∀ a ∈ R 1 is the identity element for multiplication on R Subtraction e is the identity of * if a * e = e * a = a i.e. a – e = e – a = a There is no possible value of e where a – e = e – a So, subtraction has no identity element in R Division e is the identity of * if a * e = e * a = a i.e. 𝑎﷮𝑒﷯ = 𝑒﷮𝑎﷯ = a There is no possible value of e where 𝑎﷮𝑒﷯ = 𝑒﷮𝑎﷯ = a So, division has no identity element in R*

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