Number of Commutative Binary Operations

A binary operation on a set \(A\) is a function that takes two elements from \(A\) and returns an element of \(A\). It can be represented as \(f: A \times A \to A\).

For the given set \(A = \{1, 2\}\), the number of elements in the set is \(|A| = 2\).

Understanding Binary Operations on Set A={1,2}

To define a binary operation on \(A\), we need to specify the result for every possible ordered pair from \(A \times A\).

The Cartesian product \(A \times A\) for \(A = \{1, 2\}\) consists of the following ordered pairs:

• \((1, 1)\)
• \((1, 2)\)
• \((2, 1)\)
• \((2, 2)\)

There are \(|A|^2 = 2^2 = 4\) distinct ordered pairs in \(A \times A\). For each of these 4 ordered pairs, the binary operation can map it to any of the \(|A| = 2\) elements in set \(A\).

Therefore, the total number of possible binary operations on set \(A\) is \(|A|^{|A|^2} = 2^{2^2} = 2^4 = 16\).

Condition for Commutative Binary Operation

A binary operation \(f\) is defined as commutative if for all elements \(a, b \in A\), the condition \(f(a, b) = f(b, a)\) holds true.

Let's apply this commutativity condition to the ordered pairs in our set \(A = \{1, 2\}\):

• For the pair \((1, 1)\): The condition \(f(1, 1) = f(1, 1)\) is always satisfied, meaning we can choose the value of \(f(1, 1)\) freely.
• For the pairs \((1, 2)\) and \((2, 1)\): The condition \(f(1, 2) = f(2, 1)\) must be met. This implies that the value assigned to the pair \((1, 2)\) must be identical to the value assigned to the pair \((2, 1)\).
• For the pair \((2, 2)\): The condition \(f(2, 2) = f(2, 2)\) is always satisfied, meaning we can choose the value of \(f(2, 2)\) freely.

Calculating the Number of Commutative Binary Operations

Considering the commutativity constraint, we need to make independent choices for the following mappings:

• Mapping for \((1, 1)\): The result \(f(1, 1)\) can be either 1 or 2. This gives us 2 choices.
• Mapping for \((1, 2)\): The result \(f(1, 2)\) can be either 1 or 2. This gives us 2 choices. Since \(f(1, 2)\) must be equal to \(f(2, 1)\) due to commutativity, the choice for \(f(2, 1)\) is automatically determined by the choice made for \(f(1, 2)\).
• Mapping for \((2, 2)\): The result \(f(2, 2)\) can be either 1 or 2. This gives us 2 choices.

To find the total number of commutative binary operations, we multiply the number of choices for each independent mapping:

\( \text{Number of commutative operations} = (\text{choices for } f(1,1)) \times (\text{choices for } f(1,2)) \times (\text{choices for } f(2,2)) \) \( = 2 \times 2 \times 2 = 8 \)

General Formula for Commutative Binary Operations

For a finite set \(A\) containing \(n\) elements, the number of distinct commutative binary operations that can be defined on \(A\) is given by the formula:

\( n^{n(n+1)/2} \)

In this problem, the set \(A\) has \(n = |A| = 2\) elements. Substituting \(n=2\) into the formula:

\( \text{Number of commutative operations} = 2^{2(2+1)/2} \) \( = 2^{2(3)/2} \) \( = 2^{6/2} \) \( = 2^3 \) \( = 8 \)

Both methods consistently show that there are 8 commutative binary operations on the set \(A = \{1, 2\}\).