Binary operations

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).

Step 1: Put (a=2) and (b=5) in the rule. Step 2: (2*5=2+5+2\cdot 5=17). Step 3: In such questions, do not miss the product term.

Step 1: The rule first requires the square of the first element. Step 2: (3*4=3^2+2(4)=9+8=17). Step 3: Complete powers and multiplication before addition.

Step 1: First find (5*2=2(5)-2=8). Step 2: Then (8*3=2(8)-3=13). Step 3: Always perform the operation inside the brackets first.

Step 1: First (2*3=2(2)-3=1). Step 2: Now (5*1=2(5)-1=9). Step 3: Write every intermediate value clearly to avoid option confusion.

Step 1: For identity (e), (\max(a,e)=a) must hold for every (a\in A). Step 2: (0) is the smallest element, so (\max(a,0)=a). Step 3: For a maximum operation, the smallest element is the identity.

Step 1: We need (\min(a,e)=a) for every (a\in A). Step 2: (3) is the greatest element, so (\min(a,3)=a). Step 3: For a minimum operation, the greatest element is the identity.

Step 1: For identity, write (a*e=a). Step 2: From (a+e-3=a), we get (e=3). Step 3: Cancel (a) from both sides and solve using the constant term.

Step 1: For inverse (x), (7*x=3) must hold. Step 2: (7+x-3=3) gives (x=-1). Step 3: While finding inverse, set the result equal to the identity element.

Step 1: Put (a*e=a), so (a+e+ae=a). Step 2: This gives (e(1+a)=0), and (e=0) works for every (a). Step 3: Quickly check the value in (e*a) also.

Step 1: For inverse (x), (2*x=0) must hold. Step 2: (2+x+2x=0) gives (3x=-2), so (x=-\frac{2}{3}). Step 3: Bring all terms to one side and solve a simple equation.