Question

5. Prove or disprove the following statements:

(a) Let R be a relation on the set Z of integers such that xRy if and only if xy ≥ 1. Then, R is irreflexive.

(b) Let R be a relation on the set Z of integers such that xRy if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.

(c) Let R and S be reflexive relations on a set A. Then, R − S is irreflexive.

Answer #1

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 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...

Let Z be the set of integers. Define ~ to be a relation on Z by
x~y if and only if |xy|=1. Show that ~ is symmetric and transitive,
but is neither reflexvie nor antisymmetric.

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 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
A be the set of all integers, and let R be the relation "m divides
n." Determine whether or not the given relation R, on the set A, is
reflexive, symmetric, antisymmetric, or transitive.

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 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 H be a reflexive relation on A. Prove that all relation R on
A. It is true that R ⊆ H ◦ R and R ⊆ R ◦ H.

Let R be a relation on A. Suppose that dom(R) = A and
R^(-1)∘R⊆R. Prove that R is reflexive on A.

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