Question

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

Answer #1

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.

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

Given a preorder R on a set A, prove that there is an
equivalence relation S on A and a partial ordering ≤ on A/S such
that [a] S ≤ [b] S ⇐⇒ aRb.

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.

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...

Prove that the relation R on the set of all people, defined by
xRy if x and y have the same first name is an equivalence
relation.

Suppose R is an equivalence relation on a finite set A, and
every equivalence class has the same cardinality m. Express |R| in
terms of m and |A|.
Explain why the answer is m|A|

On set R2 define ∼ by writing(a,b)∼(u,v)⇔ 2a−b =
2u−v. Prove that∼is an equivalence relation on R2
In the previous problem:
(1) Describe [(1,1)]∼. (That is formulate a statement P(x,y)
such that [(1,1)]∼ = {(x,y) ∈ R2 | P(x,y)}.)
(2) Describe [(a, b)]∼ for any given point (a, b).
(3) Plot sets [(1,1)]∼ and [(0,0)]∼ in R2.

Let ~ be an equivalence relation on a given set A. Show [a] =
[b] if and only if a ~ b, for all a,b exists in A.

Consider the following relation ∼ on the set of integers
a ∼ b ⇐⇒ b 2 − a 2 is divisible by 3
Prove that this is an equivalence relation. List all equivalence
classes.

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