Understanding Binary Operations: Definition and Properties

Binary operation (on a set G)

for all a,b in G, a*b in G

A binary operation on a set G is a mathematical operation that combines two elements from the set G to produce a third element of G. Typically, the binary operation is denoted by a symbol like •, +, ×, ∘, etc.

Formally, let G be a set, and • be a binary operation on G. Then, for any two elements a, b ∈ G, the binary operation • produces another element in G, denoted by a•b or (a,b)•. Note that the order of applying the binary operation is important, i.e., a•b may not be equal to b•a.

For a binary operation to be considered well-defined, it must satisfy certain conditions, such as closure, associativity, and existence of identity and inverses. Let’s briefly discuss these conditions:

1. Closure: A binary operation is said to be closed if and only if the result of the operation between any two elements in G is also in G. In other words, a•b ∈ G for any a, b ∈ G.

2. Associativity: A binary operation is said to be associative if and only if the way of grouping any three elements in G does not affect the result of the operation. In other words, (a•b)•c = a•(b•c) for any a, b, c ∈ G.

3. Identity: A binary operation is said to have an identity element if and only if there exists an element e ∈ G such that for any a ∈ G, e•a = a•e = a. In other words, e is the do-nothing element that leaves any element in G unchanged when the binary operation is performed.

4. Inverse: A binary operation is said to have inverse elements if and only if for any a ∈ G, there exists an element b ∈ G such that a•b = b•a = e, where e is the identity element (if it exists). In other words, the inverse element undoes the binary operation on a element, returning it back to its original state.

Examples of binary operations include addition (+) and multiplication (×) on the set of real numbers, composition (∘) of functions on a set of functions, and matrix multiplication (•) on the set of square matrices of a fixed size.

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