Binary operations

Step 1: The identity element is (0). Step 2: Put (3*x=0). Then (3+x+3x=0), so (3+4x=0). Step 3: Hence (x=-\frac{3}{4}); solve the linear equation carefully.

Step 1: The identity is (0). Step 2: Put (3*x=0). Then (3+x-3x=0), so (3-2x=0). Step 3: Hence (x=\frac{3}{2}); be careful with signs.

Step 1: For (a*e=a), we need (ae=a). Step 2: Since (a>0), (e=1). Step 3: In ordinary multiplication, the identity element is (1).

Step 1: The identity for multiplication is (1). Step 2: (5*x=1) gives (5x=1), so (x=\frac{1}{5}). Step 3: For multiplicative inverse, take the reciprocal.

Step 1: The identity for multiplication is (1). Step 2: (2*x=1) gives (x=\frac{1}{2}), which is not an integer. Step 3: The inverse must belong to the same set.

Step 1: The identity element is (0). Step 2: From (a*x=0), (a+x+ax=0), so (x=-\frac{a}{1+a}). Step 3: When (a=-1), the denominator becomes (0), so the inverse does not exist.

Step 1: The identity element is (0). Step 2: From (a*x=0), (a+x-ax=0), so (x=-\frac{a}{1-a}). Step 3: When (a=1), the denominator becomes (0), so the inverse does not exist.

Step 1: For identity (e), we need (\gcd(a,e)=a). Step 2: This happens when (e) is a multiple of every (a), and (8) works in the set. Step 3: For a (\gcd) operation on divisors of a number, the identity is often the largest element.

Step 1: For identity (e), we need (\operatorname{lcm}(a,e)=a). Step 2: (\operatorname{lcm}(a,1)=a) for every (a \in A). Step 3: For an (\operatorname{lcm}) operation, checking (1) first is useful.

Step 1: (1*2=2\cdot1+2=4), so ((1*2)*3=4*3=11). Step 2: (2*3=7), so (1*(2*3)=1*7=9). Step 3: If changing brackets changes the value, the operation is not associative.

Step 1: For identity (e), we need (a*e=a) for every (a \in A). Step 2: Here (a*e=\min(a,e)), so (e) must be the greatest element of the set. Step 3: In a minimum operation on a finite ordered set, the greatest element works as the identity.

Step 1: For identity (e), (\max(a,e)=a) must hold for all (a \in A). Step 2: This is possible only when (e) is the least element of (A). Step 3: In a maximum operation, the least element is the identity.

Step 1: Let the identity be (e). Then (a*e=a). Step 2: From (a+e+ae=a), we get (e(1+a)=0), so (e=0) works for all (a). Step 3: To find identity, use a general element and form the equation.

Step 1: First find the identity. From (a*e=a), (a+e-2=a), so (e=2). Step 2: Put (3*b=2). Then (3+b-2=2), so (b=1). Step 3: Always find the identity before finding an inverse.

Step 1: (a*b=a+b+1=b+a+1=b*a), so the operation is commutative. Step 2: ((a*b)*c=a+b+c+2) and (a*(b*c)=a+b+c+2), so it is associative. Step 3: Check commutativity and associativity separately.

Step 1: (a*b=a-b) and (b*a=b-a), which are not equal in general. Step 2: ((a*b)*c=a-b-c) but (a*(b*c)=a-b+c), so it is not associative. Step 3: For subtraction-type operations, order matters a lot.

Step 1: For identity (e), we need (a*e=a). Step 2: (\frac{ae}{2}=a) gives (e=2). Step 3: While simplifying, note that (a) is positive and hence non-zero.

Step 1: For commutativity, check whether (a*b=b*a). Step 2: (a*b=a^2+b^2) and (b*a=b^2+a^2), which are equal. Step 3: The order of terms in addition does not change the value.

Step 1: For commutativity, (a+2b=b+2a) must hold. Step 2: Simplifying gives (a=b). This is not true for all pairs. Step 3: Equality for special pairs does not make the whole operation commutative.

Step 1: The operation is based on the remainder modulo (3). Step 2: The remainder of (a+0) remains (a), so (a*0=a). Step 3: For addition modulo a number, (0) is usually the identity.