The four elementary arithmetic operations are:
Addition (+): a + b = c
Subtraction (−): a − b = c
Multiplication (×): a × b = c
Division (÷): a ÷ b = c
For any real numbers a, b, and c, these operations satisfy:
a + b = b + a (commutativity of addition)
(a + b) + c = a + (b + c) (associativity of addition)
A group ⟨G, ∗⟩ satisfies:
1. Closure: ∀a, b ∈ G, a ∗ b ∈ G
2. Associativity: ∀a, b, c ∈ G, (a ∗ b) ∗ c = a ∗ (b ∗ c)
3. Identity element: ∃e ∈ G such that ∀a ∈ G, e ∗ a = a ∗ e = a
4. Inverse elements: ∀a ∈ G, ∃b ∈ G such that a ∗ b = b ∗ a = e