Define the identity element.

Here are the solutions to your problems. 1. Find the identity for the binary operation on the set R, given that x y = xy. Step 1: Define the identity element. An element e R is the identity element for the operation if for all x R, x e = x and e * x = x. Step 2: Apply the definition to the given operation. Given the operation x y = xy, we set x e = x: xe = x Step 3: Solve for e. To solve for e, we rearrange the equation: xe - x = 0 x(e - 1) = 0 For this equation to hold true for all x R (specifically for x ≠ 0), the term (e - 1) must be zero. e - 1 = 0 e = 1 We can verify this with e x = x: 1 x = 1x = x, which is correct. The identity element is 1. 2. A binary operation is defined as a b = a + b - 2ab. i) Find the inverse element under the operation . Step 1: Find the identity element e for the operation . An element e R is the identity element if for all a R, a e = a and e a = a. Using a e = a: a + e - 2ae = a Subtract a from both sides: e - 2ae = 0 Factor out e: e(1 - 2a) = 0 For this equation to hold for all a R, e must be 0. (If e ≠ 0, then 1 - 2a = 0 a = (1)/(2), which means e would only work for a specific a, not all a.) We can verify with e a = a: 0 a = 0 + a - 2(0)a = a, which is correct. So, the identity element is e = 0. Step 2: Define the inverse element. An element a' R is the inverse of a R if a a' = e and a' a = e. We found the identity element e = 0. So, we need to solve a a' = 0. Step 3: Apply the definition and solve for a'. Using the operation a a' = a + a' - 2aa', we set it equal to 0: a + a' - 2aa' = 0 Rearrange the terms to isolate a': a' - 2aa' = -a Factor out a' from the left side: a'(1 - 2a) = -a Divide by (1 - 2a) to solve for a': a' = (-a)/(1 - 2a) a' = (a)/(2a - 1) The inverse element a' exists for all a R such that 2a - 1 ≠ 0, i.e., a ≠ (1)/(2). The inverse element is (a)/(2a - 1), for a ≠ (1)/(2). ii) The inverse of H. Step 1: Use the formula for the inverse found in part i). The inverse of an element a is given by a' = (a)/(2a - 1). Step 2: Substitute H for a. To find the inverse of H, we replace a with H in the formula: Inverse of H = (H)/(2H - 1) This inverse exists for H R such that H ≠ (1)/(2). The inverse of H is (H)/(2H - 1), for H ≠ (1)/(2). Send me the next one 📸