Binary operations

Binary operations are operations that combine two elements to produce a third element in a given set.
Operations like addition and multiplication in arithmetic are common examples.
In the context of group theory, a binary operation is defined on the set that constitutes the group.
The two inputs and the output for the operation all lie within the same set.

General Properties of Binary Operations

Closure: For all elements a, b in the set, the result of the operation a*b is also in the set.
Associative Property: For all elements a, b, and c in the set, (ab)c is equal to a(bc).
Identity Element: There exists an element, e, in the set such that for all elements a in the set, the operations ea and ae each return a.
Inverse Element: For each element a in the set, there exists an element b (the inverse of a) in the set such that the operations ab and ba each return the identity element.

In the context of group theory, binary operations must satisfy specific properties to ensure the set together with the operation forms a group.
The operation defined on a group is, by definition, closed. This means when you conduct the operation on any two elements of the group, the result is also an element within the same group.
Group operations must be associative - the way in which elements are grouped in the operation does not change the outcome.
Every group has an identity element. This is an element that, when used in the operation with any other element of the group, does not change that element.
Finally, every element in a group has an inverse. This means for any given element, there is another element in the group that, when the operation is performed with the given element, results in the identity element.

Addition (+) and multiplication (x) are binary operations on the set of real numbers, as they always produce another real number and are associative.
Subtraction (-) is not a valid binary operation in the set of natural numbers, as it does not satisfy the closure property. This is because subtracting one natural number from another can result in a negative number, which is outside of the set of natural numbers.