Page 15 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes
Example: Let S be the set of straight lines in R × R. Consider the relation on S given by
‘L R L iff L = L or L is parallel to L . Show that R is an equivalence relation, What are the
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equivalence classes in S?
Solution: R is reflexive, symmetric and transitive. Thus, R is an equivalence relation. Now, take
any line L (see Figure 1.1).
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Figure 1.1: The Equivalence Class of L 1

Let L be the line through (0, 0) and parallel to L . Then L  [L ]. Thus, [L] = [L,]. In this way the
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distinct lines through (0, 0) give distinct equivalence classes into which S is partitioned. Each
equivalence class [L] consists of all the lines in the plane that are parallel to L.
In the next section we will briefly discuss a concept that you may be familiar with, namely,
functions.

1.4 Functions

Recall that a function f from a non-empty set A to a non-empty set B is a rule which associates
with every element of A exactly one element of B. This is written as f : A  B. If f associates with
a  A, the element b of B, we write f(a) = b. A is called the domain of f, and the set f(A) = { f(a) |
a  A] is called the range of f. The range of f is a subset of B.
i.e., f(A)  B. B is called the codomain of f.
Note that

(i) For each element of A, we associate some element of B.
(ii) For each element of A, we associate only one element of B.
(iii) Two or more elements of A could be associated with the same element of B.
For example, let A = { l, 2, 3 }, B = { l, 2, 3, 4, 5, 6, 7, 8, 9, l0 }. Define f : A B by f(1) = 1, f(2) = 4,
f(3) = 9. Then f is a function with domain A and range {l, 4, 9}. In this case we can also write
f(x) = x for each x  A or f : A  B : f(x) = x . We will often use this notation for defining any
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function.

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