Equivalent Relations
A common task is to collect objects based on a common property
they share.
The equivalent symbol ≡ is used to denote that two objects
share a common property.
Objects can share a common, be equivalent (≡), without being
equal (=).
Equivalent Relations
Ask yourself, what could you say about objects that share a common property?
Let a, b, and c be objects in some universe U.
• Each object has its own properties: a ≡ a, b ≡ b, c ≡ c. (equivalence is reflexive).
• If a has b’s properties, then b has a’s properties: if a ≡ b, then
b ≡ a (equivalence is symmetric).
• If a has b’s properties, and b has c’s properties, then a has c’s
properties: if a ≡ b and b ≡ c, then a ≡ c (equivalence is transitive).
Equivalent Relations
Definition 1 (Equivalent Relation). A relation ≡ is an equivalence relation if
1. ≡ is reflexive: (∀ a)( a ≡ a)
2. ≡ is symmetric: (∀ a, b)(( a ≡ b) → (b ≡ a))
3. ≡ is transitive: (∀ a, b, c)(( a ≡ b) ∧ (b ≡ c) → ( a ≡ c))
These rules are sufficient to partition a set into disjoint equivalence
classes.
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Equality is an Equivalent Relations
Equality is an equivalence relation:
1. a = a,
2. if a = b, then b = a,
3. if a = b and b = c, then a = c.
Equality partitions the universe into singleton sets. For instance,
on the natural numbers N the equivalence classes are
{0} , {1} , {2} , {3} , {4} , {5} , . . .
Congruence Mod n is an Equivalence Relation
Congruence mod n is an equivalence relation:
1. a − a is a multiple of n,
2. if a − b = kn, then b − a = (−k)n,
3. if a − b and b − c are both multiples of n, then a − c = ( a − b) +
(b − c) is a multiple of n.
Congruence mod n partitions the integers into n equivalence classes.
For instance, congruence mod 3 partitions the integers into 3
equivalence classes. equivalence classes
1. {0, ±3, ±6, ±9, ±12 . . .}
2. {1 ± 3, 1 ± 6, 1 ± 9, ±12 . . .}
3. {2 ± 3, 2 ± 6, 2 ± 9, ±12 . . .}
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Congruence Mod n is an Equivalence Relation
The adjacency matrix for congruence mod 3 on the digits is shown
below.
Congruence mod 3 on D
0 3 6 9 1 4 7 2 5 8
0
3
6
9
1
4
7
2
5
8
1
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
0
0
0
0
0
0
0
1
1
1
Equal Fractions Define an Equivalence Relation
Let a = ( x, y), b = (u, v), and c = (w, z) be points in the punctured
two dimensional Cartesian plane. (“punctured” means the origin (0, 0)
is excluded)
y
c
b
x
a
Say two points a = ( x, y) and b = (u, v) are equivalent if
more generally,
a ≡ b if xv = yu.
y
x
=
v
u,
or
Visualize this as: two points are equivalent if they lie on the same
line through the origin.
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Equal Fractions Define an Equivalence Relation
You can show equal fractions (or equal points in projective space)
is an equivalence relation.
1. Reflexive: a fraction is equal to itself ( x, y) ≡ ( x, y) since xy =
yx.
2. Symmetric: if fraction a = ( x, y) is is equal to fraction b =
(u, v), then fraction b is is equal to fraction a
if xv = yu, then uy = vx.
Equal Fractions Define an Equivalence Relation
3. Transitive: if fraction a = ( x, y) is is equal to fraction b = (u, v)
and fraction b = (u, v) is is equal to fraction c = (w, z), then
fraction a is is equal to fraction c
yu vw = yw.
if xv = yu and uz = vw, then xz =
v
u
You can work through the special cases when one of u or v is
0.
If u = 0, then x = 0 and w = 0 (since v cannot be 0 in the
punctured plane). Therefore, (0, y) ≡ (0, z) still holds True.
Problems on Equivalence Relations
Show your understanding of this topic by completing the problems found at Equivalence Relations
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