Question

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.

Answer #1

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 and B have the same number of elements.
(a) Prove that this is an equivalence relation.
b) Determine the equivalence classes.
c) Determine the number of elements in each equivalence
class.

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
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 = {1,2,3,4,5} and X = P(A) be its powerset. Define a
binary relation on X by for any sets S, T ∈ X, S∼T if and only if S
⊆ T.
(a) Is this relation reflexive?
(b) Is this relation symmetric or antisymmetric?
(c) Is this relation 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

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.

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.

2.
Let A, B, C be subsets of a universe U.
Let R ⊆ A × A and S ⊆ A × A be binary relations on
A.
i. If R is transitive, then R−1 is transitive.
ii. If R is reflexive or S is reflexive, then R ∪ S is
reflexive.
iii. If R is a function, then S ◦ R is a function.
iv. If S ◦ R is a function, then R is a function

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