Question

Let S1 and S2 be any two equivalence relations on some set A, where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.

Prove or disprove (all three):

The relation S defined by S=S1∪S2 is

(a) reflexive

(b) symmetric

(c) transitive

Answer #1

Let S1 and S2 be any two equivalence relations on some set A,
where A ≠ ∅. Recall that S1 and S2 are each a subset of A×A.
Prove or disprove (all three):
The relation S defined by S=S1∪S2 is
(a) reflexive
(b) symmetric
(c) transitive

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

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 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.]

For each of the following relations on the set of all integers,
determine whether the relation is reflexive, symmetric, and/or
transitive:
(?, ?) ∈ ? if and only if ? < ?.
(?, ?) ∈ ? if and only ?? ≥ 1.
(?, ?) ∈ ? if and only ? = −?.
(?, ?) ∈ ? if and only ? = |?|.

6. Let S be a finite set and let P(S) denote the set of all
subsets of S. Define a relation on P(S) by declaring that two
subsets A and B are related if A ⊆ B.
(a) Is this relation reflexive? Explain your
reasoning.
(b) Is this relation symmetric? Explain your
reasoning.
(c) Is this relation transitive? Explain your
reasoning.

For Problems #5 – #9, you willl either be asked to prove a
statement or disprove a statement, or decide if a statement is true
or false, then prove or disprove the statement. Prove statements
using only the definitions. DO NOT use any set identities or any
prior results whatsoever. Disprove false statements by giving
counterexample and explaining precisely why your counterexample
disproves the claim.
*********************************************************************************************************
(5) (12pts) Consider the < relation defined on R as usual, where
x <...

Disprove: The following relation R on set Q is either reflexive,
symmetric, or transitive. Let t and z be elements of Q. then t R z
if and only if t = (z+1) * n for some integer n.

Let us say that two integers are near to one another provided
their difference is 2 or smaller (i.e., the numbers are at most 2
apart). For example, 3 is near to 5, 10 is near to 9, but 4 is not
near to 8. Let R stand for this is-near-to relation. (a) Write down
R as a set of ordered pairs. Your answer should look like this: R =
{(x, y) : . . .}. (b) Prove or disprove:...

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

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