Binary operations

Step 1: Writing (a*e=a) gives (a+e+2ae=a). Step 2: (e(1+2a)=0), so (e=0) works for every (a). Step 3: An identity element must work for every (a), not just one value.

Step 1: For inverse (x), we need (1*x=0). Step 2: (1+x+2x=0) gives (3x=-1), so (x=-\frac{1}{3}). Step 3: Combine like terms and solve the equation carefully.

Step 1: The sum of two integers is an integer. Step 2: Adding (1) still gives an integer. Step 3: For closure, only check whether the result stays in the same set.

Step 1: Negative numbers are not natural numbers. Step 2: (2*5=2-5=-3), which is not a natural number. Step 3: One valid counterexample is enough to disprove closure.

Step 1: The sum of two rational numbers is rational. Step 2: Dividing a rational number by (2) also gives a rational number. Step 3: An average-based rule remains closed on rational numbers.

Step 1: (a*b=\frac{a+b}{2}) and (b*a=\frac{b+a}{2}). Step 2: Since (a+b=b+a), both are equal. Step 3: For commutativity, compare the values in both orders.

Step 1: (1*3=2), so ((1*3)*5=2*5=\frac{7}{2}). Step 2: (3*5=4), so (1*(3*5)=1*4=\frac{5}{2}). Step 3: The two values are different, so the operation is generally not associative.

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

Step 1: For inverse (x), write (3*x=0). Step 2: (3+x-3x=0) gives (3-2x=0), so (x=\frac{3}{2}). Step 3: Handle negative signs carefully.

Step 1: (a*b=a+b-ab) and (b*a=b+a-ba). Step 2: Addition and multiplication do not change when order is reversed. Step 3: Simplify algebraic forms to check commutativity.

Step 1: (b*a=b+a+ba). Step 2: Since (b+a=a+b) and (ba=ab), both are equal. Step 3: To prove commutativity, write both forms side by side.

Step 1: (1*2=1-2+2=1). Step 2: (2*1=2-1+2=3), so they are not equal. Step 3: One unequal pair is enough to disprove commutativity.

Step 1: First (1*2=1+2+2=5). Step 2: Then (5*3=5+3+2=10). Step 3: Apply the operation twice according to the brackets.

Step 1: First (2*3=2+3+2=7). Step 2: Then (1*7=1+7+2=10). Step 3: With this rule, the brackets still give the same value here.

Step 1: ((a*b)*c=(a+b+2)+c+2=a+b+c+4). Step 2: (a*(b*c)=a+(b+c+2)+2=a+b+c+4). Step 3: If both forms are equal, associativity holds.

Step 1: For (a*e=a), we need (a+e+2=a). Step 2: This gives (e=-2), and it also works in (e*a). Step 3: The identity often looks like the opposite of the constant term.

Step 1: For inverse (x), write (5*x=-2). Step 2: (5+x+2=-2) gives (x=-9). Step 3: In inverse questions, the target value is the identity element, not necessarily zero.

Step 1: ((a+1)(b+1)=ab+a+b+1). Step 2: Subtracting (1) gives (ab+a+b). Step 3: In transformation questions, expand and match the terms.

Step 1: (x*2=x+2+2x). Step 2: (3x+2=8) gives (3x=6), so (x=2). Step 3: Convert the operation into a simple equation first.

Step 1: (x*3=2x+3). Step 2: (2x+3=11) gives (x=4). Step 3: Set the given result equal and solve the linear equation.