9.2 Give 3 examples of equivalence relations and describe the
equivalence classes.
9.3 Let R be an equivalence relation on a set S. Prove that two
equivalence classes are either equal or do not intersect. Conclude
that S is a disjoint union of all equivalence classes.
Let S1 and S2 be any two equivalence relations on some set A,
where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.
Prove or disprove (all three):
The relation S defined by S=S1∪S2 is
(a) reflexive
(b) symmetric
(c) transitive
Let X be the set of equivalence classes. So X = {[(a,b)] : a ∈
Z,b ∈ N} (recall that [(a,b)] = {(c,d) ∈Z×N : (a,b) ∼ (c,d)}).
We define an addition and a multiplication on X as follows:
[(a,b)] + [(c,d)] = [(ad + bc,bd)] and [(a,b)]·[(c,d)] =
[(ac,bd)]
Prove that this addition and multiplication is well-defined on
X.
Let f(n,k) be the number of equivalence relations with k classes
on set with n elements.
a) What is f(2,4)?
b) what is f(4,2)?
c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k *
f(n-1,k)
Show that the given relation R is an equivalence relation on set
S. Then describe the equivalence class containing the given element
z in S, and determine the number of distinct equivalence classes of
R.
Let S be the set of all possible strings of 3 or 4 letters, let
z = ABCD and define x R y to mean that x has the same first letter
as y and also the same third letter as y.
For the following relations on S = {0,1,2,3}, specify which of
the properties (R), (AR), (AS), and (T) the relations satisfy.
(R) = reflexive
(AR) = anti reflexive
(AS) = anti symmetric
(T) = transitive
b.) (m,n) in the domain of R2 if m - n is even
c.) (m,n) in the domain of R3 if m less than or equal to n
d.) (m,n) in the domain of R4 if m + n is less than or equal to...