Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 1: Equivalence relations on a set.
Given a set A, consider the set E (A) of all equivalence relations on A.
(a) Use the set A = {a, b, c, d}, Show E (A) as a set of sets of ordered pairs.
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Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 1: Equivalence relations on a set.
(b) Using the same example as in part (a) show the corresponding set of partitions of A.
2

Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 2: Ordering of Equivalence relations on a set.
(a) Given a set A, consider the relation <
E
on E (A) defined as follows: for e
1
, e
2
in E (A), e
1
<
E
e
2
if and only if for all x, y in A, x e
1
y  x e
2
y.
Prove that <
E
is a partial ordering.
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Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 2: Ordering of Equivalence relations on a set.
(b) Describe the relation <
E
given in part 2(a) in terms of the representation of E (A) as a set of sets of ordered pairs.
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Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 3: Lattice of Equivalence relations on a set.
(a) Given the poset ( E (A), <
E
), where <
E
is defined in part 2 above, prove that this poset is a
Lattice. Note, you may find it helpful to use the representation of E (A) as a set of sets of ordered pairs and the mapping of <
E
to this representation.
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Computer Science 290 Name________________________________
Lab 7, Lattices of Equivalence Relations
Part 3: Lattice of Equivalence relations on a set.
(b) For the example A = {a, b, c, d}, draw the Hasse diagram for the Lattice ( E (A), <
E
). It might be easiest to represent each equivalence relation, e, by its associated quotient, A/e, (a partition of
A) for this diagram.
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