Groups
A group builds upon the concept of a set by adding a binary operation to it. We denote a group by putting the set and the operation in angle brackets (⟨⟩). For example, ⟨A, ֎⟩ for set A and operation ֎. The operation has to follow certain rules to be considered a group, namely, the rules of identity, associativity, invertibility, and closure. If the operation ֎ also has the property of commutativity, then it is called an Abelian group (also known as a commutative group).
In our example set of mammals, the operation of sexual reproduction would not make it a group because the only property it has is commutativity.
Now, let's look at a mathematical example. What if we define ֎ to be addition over the natural numbers ℕ denoted ⟨ℕ, ֎⟩ – is this a group? Well, let's go through the properties and see if it fulfills each one.
- Identity: Does there exist an identity element...
