Oct 25, 2023

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Oct 25, 2023

1. Closure: A binary operation on a set is said to be closed if the operation always produces a result that is also in the set. In other words, for any two elements a and b in the set, the operation a * b is also in the set.

2. Associativity: A binary operation is said to be associative if the order in which the operations are performed does not affect the result. In other words, for any three elements a, b, and c in the set, (a * b) * c = a * (b * c).

3. Identity element: A binary operation has an identity element if there exists an element e in the set such that for any element a in the set, a * e = e * a = a. The identity element is unique for a given binary operation.

4. Inverse element: A binary operation has an inverse element if for every element a in the set, there exists an element b in the set such that a * b = b * a = e, where e is the identity element. The inverse element is unique for a given element.

5. Commutativity: A binary operation is said to be commutative if the order of the elements does not affect the result. In other words, for any two elements a and b in the set, a * b = b * a.

These properties are important in algebraic structures such as groups, rings, and fields, where binary operations play a central role.

2. Associativity: A binary operation is said to be associative if the order in which the operations are performed does not affect the result. In other words, for any three elements a, b, and c in the set, (a * b) * c = a * (b * c).

3. Identity element: A binary operation has an identity element if there exists an element e in the set such that for any element a in the set, a * e = e * a = a. The identity element is unique for a given binary operation.

4. Inverse element: A binary operation has an inverse element if for every element a in the set, there exists an element b in the set such that a * b = b * a = e, where e is the identity element. The inverse element is unique for a given element.

5. Commutativity: A binary operation is said to be commutative if the order of the elements does not affect the result. In other words, for any two elements a and b in the set, a * b = b * a.

These properties are important in algebraic structures such as groups, rings, and fields, where binary operations play a central role.

dennis kipkosgei

Oct 25, 2023

All operation should be unique for commutative and associative