LECTURE 16: EQUIVALENCE RELATIONS (
§
8.18.3)
Relationships of trust depend on our willingness to look not only to our own interests, but also the
interests of others. Peter Farquharson
He that findeth his life shall lose it: and he that loseth his life for my sake shall find it. Jesus Christ
1.
Relations
In many things in life, we want to think of objects as being equal. In the eyes of the law, men and
women are equal. In terms of gender they are not, but Susie and Anne do have the same gender. In
terms of being odd or even, 1 and 3 are equivalent. In terms of being exactly equal, 1 and 3 are different,
but 1 and 1 are equal. We need a way to talk about objects being related in some way, and how to
consider when objects are equivalent in some way.
Let
A
and
B
be sets.
A
relation
R
from
A
to
B
is, at the foundation, a subset of
A
×
B
.
The
elements in this subset tell us when an element from
A
is related to an element in
B
.
Example 1.
Let
A
=
{
1
,
2
,
3
}
and
B
=
{
x, y, z
}
.
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 Spring '11
 Smith
 Equivalence relation, relation, Peter Farquharson

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