3.2.1 Preliminary Definitions and Assumptions
Knowing is not enough; we must apply.
Willing is not enough; we must do.
—Johann von Goethe (1749–1842)

Definition. A mapping (or function f from A to B) of a set A into a set B is a rule that pairs each element of A with exactly one element of a subset of B. The set A is called the domain, and the set of all elements of B (a subset of B) that are paired with an element from A is called the range.

Definition. A mapping f from A to B is onto B if for any b in B there is at least one a in A such that f(a) = b.

Definition. A mapping f from A to B is one-to-one if each element of the range of f is the image of exactly one element from A.

Definition. A transformation is a one-to-one mapping of a set A onto a set B.

Definition. A transformation of a plane is a transformation that maps points of the plane onto points in the plane.

Definition. A nonempty set G is said to form a group under a binary operation, *, if it satisfies the following conditions:

1. If A and B are in G, then A*B is in G. (The set is closed under the operation, closure.)
2. There exists an element I in G such that for every element A in G, I*A = A*I = A. (The set has an identity.)
3. For every element A in G, there is an element B in G such that A*B = B*A = I, denoted A^–1. (Every element has an inverse.)
4. If A, B, and C are in G, then (A*B)*C = A*(B*C). (associativity)

Theorem 3.0. The set of transformations of a plane is a group under composition.

Proof. The result follows from the following:
       The composition of two transformations of a plane is a transformation.
       The inverse of a transformation is a transformation.
       The identity function is a transformation and composition of functions is associative.//

Exercise 3.2. Which of the following mappings are transformations? Justify.

a.        such that .

b.       such that .

c.        such that .

d.       such that .

e.       Let P be a point in a plane S. Define  by f(P) = P and for any point , f(Q) is the midpoint of .

Exercise 3.3.  Let  and  be transformations defined respectively by f(x,y) = (x  4, y + 1) and g(x,y) = (x + 2, y + 3).

a.       Find the composition .

b.      Find the composition .

c.       Find the inverse of f,  f ¹.

d.      Find the inverse of g,  g ¹.

Exercise 3.4. Prove the composition of two transformations of a plane is a transformation of the plane.

Exercise 3.5. Prove the inverse of a transformation of a plane is a transformation of the plane.