Page 236 - HOW TO PROVE IT: A Structured Approach, Second Edition

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222 Relations
To see that C m is transitive, suppose that x ≡ y (mod m) and y ≡ z (mod
m). Then m | (x − y) and m | (y − z), so by exercise 18(a) in Section 3.3,
m | [(x − y) + (y − z)]. But (x − y) + (y − z) = x − z, so it follows that
m | (x − z), and therefore x ≡ z (mod m). For more about these equivalence
relations, see exercise 10.
Equivalence relations often come up when we want to group together ele-
ments of a set that have something in common. For example, if you’ve studied
vectors in a previous math course or perhaps in a physics course, then you may
have been told that vectors can be thought of as arrows. But you were probably
also told that different arrows that point in the same direction and have the same
length must be thought of as representing the same vector. Here’s a more lucid
explanation of the relationship between vectors and arrows. Let A be the set of
all arrows, and let R ={(x, y) ∈ A × A | the arrows x and y point in the same
direction and have the same length}. We will let you check for yourself that R
is an equivalence relation on A. Each equivalence class consists of arrows that
all have the same length and point in the same direction. We can now think
of vectors as being represented, not by arrows, but by equivalence classes of
arrows.
Students who are familiar with computer programming may be interested in
our next example. Suppose we let P be the set of all computer programs, and
for any computer programs p and q we say that p and q are equivalent if they
always produce the same output when given the same input. Let R ={(p, q) ∈
P × P | the programs p and q are equivalent}. It is not hard to check that R is
an equivalence relation on P. The equivalence classes group together programs
that produce the same output when given the same input.

Exercises

1. Find all partitions of the set A ={1, 2, 3}.
∗
2. Find all equivalence relations on the set A ={1, 2, 3}.
3. Let W = the set of all words in the English language. Which of the
∗
following relations on W are equivalence relations? For those that are
equivalence relations, what are the equivalence classes?
(a) R ={(x, y) ∈ W × W | the words x and y start with the same
letter}.
(b) S ={(x, y) ∈ W × W | the words x and y have at least one letter in
common}.
(c) T ={(x, y) ∈ W × W | the words x and y have the same number of
letters}.