Question

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.

Answer #1

Using Discrete Math
Let ρ be the relation on the set of natural numbers N given by:
for all x, y ∈ N, xρy if and only if x + y is even. Show that ρ is
an equivalence relation and determine the equivalence classes.

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.

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.

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

Let R1 and R2 be equivalence relations on a set A. (a) Must
R1∪R2 be an equivalence relation? (b) Must R1∩R2 be an equivalence
relation? (c) Must R1⊕R2 be an equivalence relation?[⊕is the
symmetric difference:x∈A⊕B if and only if x∈A,x∈B, and x
/∈A∩B.]

Let Q be the set {(a, b) ∶ a ∈ Z and b ∈ N}.
Let (a, b), (c, d) ∈ Q. Show that (a, b) ∼ (c, d) if and only if
ad − bc = 0 defines an equivalence relation on Q.

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

I have a discrete math question.
let R be a relation on the set of all real numbers
given by cry if and only if x-y = 2piK for some integer K. prove
that R is 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.

Let
X be finite set . Let R be the relation on P(X). A,B∈P(X) A R B Iff
|A|＝|B| prove R is an equivalence relation

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