Show that m = m + 0 = 0 + m, for all m belonging to naturals, that is, 0 is a neutral element of the addition operation in

Certainly! To show that 0 is the neutral element of addition for natural numbers (N), we need to demonstrate that for any natural number 'm,' the following equations hold true: 1. m + 0 = m 2. 0 + m = m Let's prove each of these equations: 1. m + 0 = m: We can prove this by induction. Base Case: For m = 1, 1 + 0 = 1, which is true. Inductive Step: Assume it's true for some natural number k: k + 0 = k We need to show that it's true for k + 1: (k + 1) + 0 = k + (1 + 0) (using associativity) (k + 1) + 0 = k + 1 So, by induction, m + 0 = m for all natural numbers m. 2. 0 + m = m: This can be proven similarly by induction. Base Case: For m = 1, 0 + 1 = 1, which is true. Inductive Step: Assume it's true for some natural number k: 0 + k = k We need to show that it's true for k + 1: 0 + (k + 1) = (0 + k) + 1 (using associativity) 0 + (k + 1) = k + 1 So, by induction, 0 + m = m for all natural numbers m. Therefore, we have shown that 0 is the neutral element of the addition operation for natural numbers, as both equations m + 0 = m and 0 + m = m hold true for all natural numbers m.