Binary operations

Step 1: For identity element (e), (a*e=a) and (e*a=a). Step 2: (a+0=a) and (0+a=a). Step 3: For addition-like operations, check (0) first.

Step 1: An identity element (e) must satisfy (ae=a) and (ea=a). Step 2: (a\cdot 1=a) and (1\cdot a=a). Step 3: For multiplication-like operations, (1) is the first candidate.

Step 1: We need (a*e=a). Step 2: From (a+e+1=a), we get (e=-1). Step 3: To find identity, cancel (a) from both sides and solve for (e).

Step 1: For (a*e=a), we need (a+e-1=a). Step 2: This gives (e=1), and (e*a=e+a-1=a) also works. Step 3: It is a good habit to check identity from both sides.

Step 1: (a*0=a+0+a\cdot 0=a). Step 2: (0*a=0+a+0\cdot a=a), so (0) works from both sides. Step 3: The identity element leaves every element unchanged.

Step 1: For inverse (x), we need (5*x=0). Step 2: (5+x=0) gives (x=-5). Step 3: In addition, the inverse is the element with the opposite sign.

Step 1: For inverse (x), we need (4x=1). Step 2: Hence (x=\frac{1}{4}). Step 3: In multiplication, think of the reciprocal as the inverse.

Step 1: For inverse (x), (3*x=1) must hold. Step 2: (3+x-1=1) gives (x=-1). Step 3: First identify the identity, then form the inverse equation.

Step 1: For inverse (x), (2*x=-1) must hold. Step 2: (2+x+1=-1) gives (x=-4). Step 3: In inverse questions, set the result equal to the identity element.

Step 1: First calculate inside the bracket: (2*3=2+3=5). Step 2: Then (5*4=5+4=9). Step 3: In bracket questions, do the inner operation first.

Step 1: First (3*4=3+4=7). Step 2: Now (2*7=2+7=9). Step 3: The operation rule remains the same, only the order follows the brackets.

Step 1: ((2*3)*4=9) and (2*(3*4)=9). Step 2: The two values are equal, showing associativity in this example. Step 3: In addition-based operations, changing brackets does not change the value.

Step 1: First (5*2=5-2=3). Step 2: Then (3*1=3-1=2). Step 3: In subtraction, bracket order can change the answer.

The governing concept is a defined binary operation together with evaluation of a nested expression. The rule a * b = a − b must be used whenever the symbol * appears; it does not mean ordinary multiplication. Because the expression contains parentheses, evaluate the inner operation first: 2 * 1 = 2 − 1 = 1. Substitute this result into the outer operation: 5 * (2 * 1) = 5 * 1 = 5 − 1 = 4. Therefore option B is correct. Option A is the result of stopping after the inner operation. Option C may arise from adding or mishandling the operands, and option D does not follow the defined rule. Correct bracket order and the given operation rule are both essential.

Step 1: ((5*2)*1=2). Step 2: (5*(2*1)=4), so the two values are not equal. Step 3: One unequal example is enough to disprove associativity.

Step 1: First (2*3=2\cdot 3=6). Step 2: Then (6*4=6\cdot 4=24). Step 3: Under multiplication, multiply the two numbers each time.

Step 1: First (3*4=3\cdot 4=12). Step 2: Then (2*12=2\cdot 12=24). Step 3: In multiplication, changing brackets still gives the same final product.

Step 1: (\max(a,b)) means choosing the greater value. Step 2: Between (3) and (7), the greater value is (7). Step 3: In a maximum-based rule, do not add or multiply the numbers.

Step 1: (\min(a,b)) means choosing the smaller value. Step 2: Between (8) and (5), the smaller value is (5). Step 3: In a minimum-based operation, simply choose the smaller element.

Step 1: (a*b=\max(a,b)) and (b*a=\max(b,a)). Step 2: The greater value is the same in both cases, so the results are equal. Step 3: Maximum and minimum operations are easy to test for commutativity.