Exercises
- Let j be a binary relation in the set Z of integer numbers
defined below
x j y iff x mod 5 = y mod 5.
- Give the arguments demonstrating that relation j is an
equivalence relation.
- Identify its classes of equivalence.
- We assume: for every n, m ∈
N, n j m iff m2 - n2
is a multiple of 3.
- Prove that ∈is an equivalence
relation.
- List a few elements of the class [0] and the class [1].
- How many abstraction classes have this relation?
- Let j be
the relation in the set Z of integer numbers defined by: n j m iff n mod 9
= m mod 9.
Prove that the number abcd (written in the decimal notation - i.e. a,
b, c, d are decimal digits) belongs to the class [0] iff the number
(a+b+c+d) belongs to class [0].
- In the power set P(X) such that x0 ∈
X we define the relation:
A j B iff
x0 ∈
A and x0 ∈ B or x0 ∉
A and x0 ∉ B
- Is j an equivalence
relation?
- Show its abstraction classes.
- Let r1 and r2 be two equivalence
classes in a set X.
Show that r1 ∪ r2 is
an equivalence relation
iff r1 ∪ r2 = r1o
r2.
- Provide an arbitrarily chosen partition of the set R × R. Define equivalence relation such that
the abstraction classes form the given partition.
- How many equivalence relations may be defined in an n element set?