Binary operations

Step 1: The entries of the table show operation results. Step 2: If every result is in the same set, closure holds. Step 3: In table-based questions, check all entries first.

Step 1: For a binary operation, every result must lie in (A). Step 2: If any result is outside (A), closure fails. Step 3: Once closure fails, it is not a binary operation on (A).

Step 1: (A\times A) has (2^2=4) ordered pairs. Step 2: Each pair has (2) possible outputs, so total operations are (2^4=16). Step 3: Use (|A|^{|A|^2}) for the number of binary operations.

Step 1: (A\times A) has (3^2=9) ordered pairs. Step 2: Each pair has (3) choices for its output. Step 3: Therefore, the number of binary operations is (3^9).

Step 1: On integers, the result must be an integer. Step 2: (1) and (2) are integers, but their average (\frac{3}{2}) is not an integer. Step 3: Always check closure in average-based rules.

Step 1: The sum of two rational numbers is rational. Step 2: Dividing a rational number by (2) again gives a rational number. Step 3: Hence it is closed on (\mathbb{Q}).

Step 1: For any real (a,b), (a^2) and (b^2) are real. Step 2: Their sum is also real. Step 3: Since the result stays in real numbers, closure holds.

Step 1: After changing the order, (b*a=b^2+a^2). Step 2: Addition is commutative, so (a^2+b^2=b^2+a^2). Step 3: Commutativity is easy to identify in addition-based forms.

Step 1: Put (a*e=a). Step 2: From (a+e+2=a), we get (e=-2). Step 3: In such rules, the negative of the added constant often becomes the identity.

Step 1: The identity of this operation is (-2). Step 2: Put (3*b=-2), so (3+b+2=-2). Step 3: This gives (b=-7), so the inverse should be (-7).

Step 1: For a binary operation, every result for (a,b\in A) must lie in (A). Step 2: Here (a\ast b=a), and (a) always belongs to (A). Step 3: In exams, first check closure of the result.

Step 1: For a binary operation on (N), (a-b) must always be in (N). Step 2: (2-5=-3), which is not in (N). Step 3: One counterexample is enough to reject closure.

Step 1: The sum of integers is always an integer. Step 2: (a+b+1) is also an integer for every (a,b\in Z). Step 3: On (Z), expressions made using addition usually remain closed.

Step 1: An identity element (e) must satisfy (a\ast e=a) and (e\ast a=a). Step 2: In multiplication, (a\cdot1=a) and (1\cdot a=a). Step 3: For multiplication, quickly identify (1) as the identity.

Step 1: The identity element for addition is (0). Step 2: The inverse of (5) is a number (x) such that (5+x=0), so (x=-5). Step 3: For addition, the inverse is the negative of the number.

Step 1: The identity is (1) because multiplication by (1) does not change a number. Step 2: (1\cdot1=1), so (1) is its own inverse. Step 3: In a small set, checking options directly is helpful.

Step 1: For identity (e), (a\ast e=a) must hold. Step 2: From (a+e-2=a), we get (e=2). Also, (e+a-2=a) gives (e=2). Step 3: Always check identity from both sides.

Step 1: Put (a=0) and (b=5) in the given rule. Step 2: (0\ast5=0+5+0\cdot5=5). Step 3: Even in simple calculations, apply the operation rule carefully.

Step 1: The rule is (a\ast b=a+2b). Step 2: (3\ast4=3+2\cdot4=11). Step 3: Do not interchange the positions of (a) and (b).

Step 1: For commutativity, (a\ast b=b\ast a) must hold. Step 2: (a\ast b=2a+b), while (b\ast a=2b+a), which are not generally equal. Step 3: A simple example like (a=1,b=2) checks this quickly.