Binary Operator

• https://math.stackexchange.com/questions/911787/clarify-cartesian-products-and-binary-operations

• https://www.youtube.com/watch?v=VzsAehzmjrU

Binary Operation

A binary operation on a set S is a function f: S x S -> S

• Takes 2 elements from the same set(could be the same element twice) to another element within the same set

• Addition on real numbers is a closed binary operation

• Division over integers is not closed because it can produce rational numbers.

Binary operator is a function that takes an ordered pair, containing 2 elements from the same set, and maps that to another element within the same set.

• https://www.youtube.com/watch?v=VzsAehzmjrU

Closure

Binary operators do not necessarily have to be closed.

1. Closed Binary Operator: A binary operator is considered "closed" if, when applied to any two elements from the set, it produces a result that also belongs to the same set. Closure ensures that the set is closed under the operation, and it's an essential property for certain algebraic structures like groups and rings.

2. Non-Closed Binary Operator: A binary operator is "non-closed" if there are cases where applying the operator to two elements from the set results in a value that is not in the same set.

While closure is an important property in algebraic structures, there are contexts where non-closed binary operators are studied as well. For example:

• In some mathematical investigations, non-closed binary operators are used to study how sets and operations behave in specific situations.

• In abstract algebra, non-closed binary operators can be studied within the framework of magma-like structures, where closure is not necessarily assumed.

Binary Operation and Modular Arithmetic

• https://www.youtube.com/watch?v=NNITlEPMS5A