Binary operations

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.

Step 1: The rule chooses the smaller value. Step 2: Between (0) and (6), the smaller value is (0). Step 3: In a minimum operation, zero is compared just like any other number.

Step 1: First find the difference: (9-4=5). Step 2: (|5|=5), so (9*4=5). Step 3: Absolute value gives a non-negative distance-like value.

Step 1: (4-9=-5). Step 2: (|-5|=5), so the answer is (5). Step 3: Because of absolute value, the reversed order also gives the same value here.

Step 1: (9*4=|9-4|=5). Step 2: (4*9=|4-9|=5), so both are equal. Step 3: When (a*b=b*a), it indicates commutativity.

Step 1: Square both elements. Step 2: (2*3=2^2+3^2=4+9=13). Step 3: In a sum of squares, calculate powers first.

Step 1: (a*b=a^2+b^2) and (b*a=b^2+a^2). Step 2: Addition does not change when terms are interchanged, so both are equal. Step 3: Reordering terms in a sum keeps the same value.

Step 1: First square both numbers. Step 2: (3*2=3^2-2^2=9-4=5). Step 3: In a square-difference rule, pay attention to order.

Step 1: The rule subtracts the square of the second element from the square of the first. Step 2: (2*3=2^2-3^2=4-9=-5). Step 3: Reversing order in subtraction-based rules can change the sign.

Step 1: (3*2=9-4=5). Step 2: (2*3=4-9=-5), so the values are not equal. Step 3: Reversing the order and comparing values is the simplest test for commutativity.

Step 1: For identity element (e), we need (a*e=a). Step 2: From (a+e-2=a), we get (e=2). Step 3: In such questions, first write the identity condition and then solve for (e).