Binary operations

Step 1: (\mathbb{Q}) contains rational numbers. Step 2: The sum of two rational numbers is again rational. Step 3: Since closure holds, it is a binary operation on (\mathbb{Q}).

Step 1: The result must again lie in (\mathbb{N}). Step 2: (1) and (2) are natural, but (\sqrt{3}) is not a natural number. Step 3: For square-root rules, check closure carefully.

Step 1: If (a,b\in\mathbb{Z}), then (a-b) is an integer. Step 2: (|a-b|) is a non-negative integer, and it belongs to (\mathbb{Z}). Step 3: (\mathbb{Z}) includes all non-negative integers too.

Step 1: Check possible products (0\cdot0,0\cdot1,1\cdot0,1\cdot1). Step 2: All results are (0) or (1). Step 3: For a small set, checking all pairs is an easy method.

Step 1: Closure must hold for every pair. Step 2: (1) and (1) are in the set, but their sum (2) is not in the set. Step 3: One failed pair breaks closure.

Step 1: This is addition using remainder modulo (2). Step 2: Doing (a*0) leaves the remainder as (a). Step 3: In remainder-based addition, (0) acts as the identity.

Step 1: The identity in this operation is (0). Step 2: In (1*1), (1+1=2), whose remainder modulo (2) is (0). Step 3: Therefore (1) is its own inverse.

Step 1: In remainder-based addition, adding (0) does not change the element. Step 2: The remainder of (a*0) is (a). Step 3: So (0) is the identity element.

Step 1: The identity element is (0). Step 2: (2+1=3), and the remainder on division by (3) is (0). Step 3: Therefore the inverse of (2) is (1).

Step 1: Here changing the order of two elements does not change the result. Step 2: This is the definition of commutativity. Step 3: Remember this property using usual addition and multiplication.

Step 1: Here changing the grouping does not change the result. Step 2: This property is called associativity. Step 3: In associativity, the order is not changed; only grouping changes.

Step 1: Operating with (e) does not change (a). Step 2: Such an element is called the identity element. Step 3: The identity should work from both sides.

Step 1: An inverse is an element that combines with (a) to give the identity. Step 2: Here (a*b=e) and (b*a=e), so (b) is the inverse. Step 3: Always understand inverse with respect to an identity element.

Step 1: (1*2=1). Step 2: (2*1=2), so the two results are not equal. Step 3: For commutativity, both orders must give the same result.

Step 1: According to the given rule, the result is the second element. Step 2: In (1*2), the second element is (2). Step 3: The operation rule may be different from usual addition or multiplication.

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).