3.2.1
Preliminary Definitions and Assumptions
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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:
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 (Exercise 3.4).
The inverse of a transformation is a transformation
(Exercise 3.5).
The identity function is a transformation and composition of functions is
associative (Exercise 3.5).//
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. (a) Prove the identity function is a transformation. (b) Prove the inverse of a transformation of a plane is a transformation of the plane. (c) Prove the composition of functions is associative.
3.1.2 Historical Overview |
© Copyright 2005, 2006 - Timothy Peil |