Understanding Binary Operations on Natural Numbers

A binary operation '*' on a set A is a rule that assigns to every ordered pair (a, b) of elements in A an element in A. Here, the set is N (natural numbers), and the operation is defined as a * b = a³ + b³. We need to determine if this operation is commutative and/or associative.

Checking for Commutativity

A binary operation '*' on a set A is said to be commutative if a * b = b * a for all elements a, b ∈ A. Let's check the given operation a * b = a³ + b³ on N. For any a, b ∈ N:

• Calculate a * b: \[a * b = a^3 + b^3\]
• Calculate b * a: \[b * a = b^3 + a^3\]

Since addition of natural numbers is commutative, we know that a³ + b³ is always equal to b³ + a³ for all a, b ∈ N. Therefore, a * b = b * a for all a, b ∈ N. The operation * is commutative.

Checking for Associativity

A binary operation '*' on a set A is said to be associative if (a * b) * c = a * (b * c) for all elements a, b, c ∈ A. Let's check the given operation a * b = a³ + b³ on N. For any a, b, c ∈ N:

• Calculate (a * b) * c: \[(a * b) * c = (a^3 + b^3) * c\] Now apply the definition of *: the first element is (a³ + b³) and the second element is c. \[(a^3 + b^3) * c = (a^3 + b^3)^3 + c^3\]
• Calculate a * (b * c): \[a * (b * c) = a * (b^3 + c^3)\] Now apply the definition of *: the first element is a and the second element is (b³ + c³). \[a * (b^3 + c^3) = a^3 + (b^3 + c^3)^3\]

For the operation to be associative, we must have $(a^3 + b^3)^3 + c^3 = a^3 + (b^3 + c^3)^3$ for all a, b, c ∈ N. Let's test with specific natural numbers, for example, a=1, b=2, c=3.

• (a * b) * c: \[(1 * 2) * 3 = (1^3 + 2^3) * 3 = (1 + 8) * 3 = 9 * 3\] \[9 * 3 = 9^3 + 3^3 = 729 + 27 = 756\]
• a * (b * c): \[1 * (2 * 3) = 1 * (2^3 + 3^3) = 1 * (8 + 27) = 1 * 35\] \[1 * 35 = 1^3 + 35^3 = 1 + 42875 = 42876\]

Since $756 \neq 42876$, the equation $(a^3 + b^3)^3 + c^3 = a^3 + (b^3 + c^3)^3$ is not true for all a, b, c ∈ N. The operation * is not associative.

Summary of Properties

Based on the analysis:

• The binary operation * defined as a * b = a³ + b³ on N is commutative.
• The binary operation * defined as a * b = a³ + b³ on N is not associative.

Therefore, the operation * is commutative but not associative.