240 RELATIONS: EQUIVALENCE RELATIONS AND PARTIAL ORDERINGS Chapter 7(6) Let F be the set of all real-valued functions having domain R. Define a rela-
tion - on F by the rule f - g if and only if {XI f(x) # g(x)} is finite. Prove that- is an equivalence relation on F. (Hint: Recall Exercise 7, Article 6.2.)
*7. Define a relation - on the set Q = Z x (Z - (0)) by the rule (m, n) - (p, q) if
and only if mq = np. Prove that - is an equivalence relation on Q.
- Give two examples, one by listing ordered pairs, the other described by a rule
(as in Example 4), of relations which are:
(a) NR, NS, T
(4 NR, S, T
(e) NR, S, NT
(6) R, NS, T
*(d) R, NS, NT
(f) NR, NS, T- Determine which of the three properties R, S, and T are possessed by each of
the following relations:
(a) L = the set of all lines in 3-space. Line 1 is related to line m if and only if
(i)^1 equals m or^1 is parallel to m (ii)^1 is perpendicular to m
(iii) 1 and m are coplanar (iv) 1 and m are skew
*(v)^1 and m intersect in a point
(6) D = the set of all triangles in a plane. Triangle X is related to triangle Y if
and only if
(i) X and Y are congruent (ii) X and Yare similar - Calculate all equivalence classes [n], where the integer n ranges from n = - 9
to n = 10, corresponding to the equivalence relation, congruence modulo 5. De-
scribe the set Z/ =, of all equivalence classes.
7.3 Equivalence Classes and Partitions
In Example 5(b) of the previous article we observed that the collection of
subsets [x] corresponding to a particular equivalence relation had a form
different from that associated with the relations from part (a) of the example
(which were not equivalence relations). In particular, we noted that the
collection AIR,, satisfies three properties, but neither of the collections
from (a) satisfies all these three properties. The next definition highlights
the three properties.DEFINITION 1 -
Let A be a set. A collection (p of subsets of A is said to be a partition of A if
and only if:(i) S #^0 for each S E 9.
(ii) If S,, S, E (p, then either S, = S, or S, n S, = 0.
(iii) For each a E A, there is some SE (p such that a E S. In symbols,
u (S~E (p) = A.1 The elements of are often referred to as cells of the partition.