Question

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

Answer #1

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.

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,...

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
A be the set of all real numbers, and let R be the relation "less
than." Determine whether or not the given relation R, on the set A,
is reflexive, symmetric, antisymmetric, or transitive.

Let A be the set of all lines in the plane. Let the relation R
be defined as:
“l1 R l2 ⬄ l1 intersects
l2.” Determine whether S is reflexive, symmetric, or
transitive. If the answer is “yes,” give a justification (full
proof is not needed); if the answer is “no” you must give a
counterexample.

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

Let A = R x R, and let a relation S be defined as: “(x1 , y1 ) S
(x2 , y2 ) ⬄ points (x1 , y1 ) and (x2 , y2 ) are 5 units apart.”
Determine whether S is reflexive, symmetric, or transitive. If the
answer is “yes,” give a justification (full proof is not needed);
if the answer is “no” you must give a counterexample

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

Suppose we define the relation R on the set of all people by the
rule "a R b if and only if a is Facebook friends with b." Is this
relation reflexive? Is is symmetric? Is
it transitive? Is it an equivalence relation?
Briefly but clearly justify your answers.

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