A set is a well-defined collection of distinct things or objects, such as animals, plants, vowels (a, e, i, o, u), swearwords, etc.
By well-defined, we mean that there is a rule that enables us to determine whether a given object is an element of the set or not.
A field is a set F together with two binary operations on F called addition and multiplication. These operations are required to satisfy the following properties, referred to as field axioms:
Examples: ℚ, ℝ, and ℂ are fields, but ℕ (2 - 3 ∉ ℕ), ℤ (3 does not have a multiplicative inverse), and ℤn are not fields. 2 and 4 do not have multiplicative inverses in ℤ6, there is no an element a ∈ ℤ6 such that a·2≡1 (mod 6).
ℤn is a field if and only if n is prime.
Fields are the environment where linear equations can be solved. Let F be a field, ax + b = c is a linear equation with variable x and a, b, c ∈ F (elements in the field are called scalars). A solution is any value or assignment to the variable x with makes the equation true or produces a true statement.
For linear equations ax + b = c, there is one and only one solution in any field. ax + b = c ⇒ [Additive inverses] (ax + b) + (-b) = c -b ⇒ [Associativity of addition] ax + (b - b) = c -b ⇒ [Additive identity] ax = c - b ⇒ [Multiplicative inverses] a-1(ax) = a-1(c -b) ⇒ [Associativity of multiplication] (a-1a)x = c-b⁄a ⇒ [Multiplicative identity] x = c-b⁄a
Solve the following equations: