Question

Let A be a non-empty set. Prove that if ∼ defines an equivalence relation on the set A, then the set of equivalence classes of ∼ form a partition of A.

Answer #1

Prove that the relation of set equivalence is an equivalence
relation.

Let A = {1, 2, 3, 4, 5}. Describe an equivalence relation R on
the set A that produces the following partition (has the sets of
the partition as its equivalence classes): A1 = {1, 4}, A2 = {2,
5}, A3 = {3} You are free to describe R as a set, as a directed
graph, or as a zero-one matrix.

Let R be an equivalence relation defined on some set A.
Prove using mathematical induction that R^n is also an
equivalence relation.

a)
Let R be an equivalence relation defined on some set A. Prove
using induction that R^n is also an equivalence relation. Note: In
order to prove transitivity, you may use the fact that R is
transitive if and only if R^n⊆R for ever positive integer n
b)
Prove or disprove that a partial order cannot have a cycle.

Let A be a non-empty set and f: A ? A be a function.
(a) Prove that, if f is injective but not surjective (which
means that the set A is infinite), then f has at least two
different left inverses.

1.
a. Consider the definition of relation. If A is the set of even
numbers and ≡ is the subset of ordered pairs (a,b) where a<b in
the usual sense, is ≡ a relation? Explain.
b. Consider the definition of partition on the
bottom of page 18. Theorem 2 says that the equivalence classes of
an equivalence relation form a partition of the set. Consider the
set ℕ with the equivalence relation ≡ defined by the rule: a≡b in ℕ...

For each of the following, prove that the relation is an
equivalence relation. Then give the information about the
equivalence classes, as specified.
a) The relation ∼ on R defined by x ∼ y iff x = y or xy = 2.
Explicitly find the equivalence classes [2], [3], [−4/5 ], and
[0]
b) The relation ∼ on R+ × R+ defined by (x, y) ∼ (u, v) iff x2v
= u2y. Explicitly find the equivalence classes [(5, 2)] and...

Let N* be the set of positive integers. The relation
∼ on N* is defined as follows: m ∼ n ⇐⇒ ∃k ∈
N* mn = k2
(a) Prove that ∼ is an equivalence relation.
(b) Find the equivalence classes of 2, 4, and 6.

Let X be a non-empty finite set with |X| = n. Prove that the
number of surjections from X to Y = {1, 2} is (2)^n− 2.

Consider the set Pn of all partitions of the set [n] into
non-empty blocks, that is:
Pn = {π | π is a set partition of [n]}
Prove that this is a poset

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