State and prove fundamental theorem of equivalence classes
Answers
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The relation "is equal to" is the canonical example of an equivalence relation, where for any objects a, b, and c:
a = a (reflexive property),
if a = b then b = a (symmetric property), and
if a = b and b = c then a = c (transitive property).
As a consequence of the reflexive, symmetric, and transitive properties, any equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class.
Simple example
Let the set {\displaystyle \{a,b,c\}} \{a,b,c\} have the equivalence relation {\displaystyle \{(a,a),(b,b),(c,c),(b,c),(c,b)\}} \{(a,a),(b,b),(c,c),(b,c),(c,b)\}. The following sets are equivalence classes of this relation:
{\displaystyle [a]=\{a\},~~~~[b]=[c]=\{b,c\}} [a]=\{a\},~~~~[b]=[c]=\{b,c\}.
The set of all equivalence classes for this relation is {\displaystyle \{\{a\},\{b,c\}\}} \{\{a\},\{b,c\}\}. This set is a partition of the set {\displaystyle \{a,b,c\}} \{a,b,c\}.
Equivalence relations
The following are all equivalence relations:
"Is equal to" on the set of numbers. For example, {\displaystyle {\tfrac {1}{2}}} {\tfrac {1}{2}} is equal to {\displaystyle {\tfrac {4}{8}}} {\displaystyle {\tfrac {4}{8}}}.
"Has the same birthday as" on the set of all people.
"Is similar to" on the set of all triangles.
"Is congruent to" on the set of all triangles.
"Is congruent to, modulo n" on the integers.
"Has the same image under a function" on the elements of the domain of the function.
"Has the same absolute value" on the set of real numbers
"Has the same cosine" on the set of all angles.