Question

For each of the following relations on the set {1, 2, 3, 4}

(a) { (1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 3), (2, 4), (3, 3), (3, 4), (4, 4) }

(b) { (1, 2), (1, 3), (1, 4), (2, 3), (2, 4), (3, 4) }

(c) { (2, 4}, (4, 2) }

(d) ( (1, 3), (1, 4), (2, 3), (2, 4), (3, 1), (3, 4) }

Choose all answers that apply.

Group of answer choices

(a) is reflexive

(d) is transitive

(d) is antisymmetric

(b) is reflexive

(a) is antisymmetric

(b) is antisymmetric

(c) is symmetric

(d) is reflexive

(a) is transitive

Answer #1

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 ? = |?|.

Consider these relations on the set of integers
R1 = { (a,b) | a < b or a ≥ b}
R2 = { (a,b) | a + b < 5 }
R3 = { (a,b) | a <= b }
R4 = { (a,b) | a = b +3 }
R5 = { (a,b) | a < b - 1 }
R6 = { (a,b) | a + 2 > b }
Choose following pairs that fit at least four...

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

Construct a binary relation R on a nonempty set A satisfying the
given condition, justify your solution.
(a) R is an equivalence relation.
(b) R is transitive, but not symmetric.
(c) R is neither symmetric nor reflexive nor transitive.
(d) (5 points) R is antisymmetric and symmetric.

Problem 3
For two relations R1 and
R2 on a set A, we define the
composition of R2 after R1
as
R2°R1 = { (x,
z) ∈ A×A | (∃ y)( (x,
y) ∈ R1 ∧ (y, z) ∈
R2 )}
Recall that the inverse of a relation R, denoted
R -1, on a set A is defined as:
R -1 = { (x, y) ∈
A×A | (y, x) ∈ R)}
Suppose R = { (1, 1), (1, 2),...

Consider the following relation on the set Z: xRy ?
x2 + y is even.
For each question below, if your answer is "yes", then prove it, if
your answer is "no", then show a counterexample.
(i) Is R reflexive?
(ii) Is R symmetric?
(iii) Is R antisymmetric?
(iv) Is R transitive?
(v) Is R an equivalence relation? If it is, then describe the
equivalence classes of R. How many equivalence classes are
there?

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...

For each of the properties reflexive, symmetric, antisymmetric,
and transitive, carry out the following.
Assume that R and S are nonempty relations on a set A that both
have the property. For each of Rc, R∪S, R∩S, and R−1, determine
whether the new relation
must also have that property;
might have that property, but might not; or
cannot have that property.
A ny time you answer Statement i or Statement iii, outline a
proof. Any time you answer Statement ii,...

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:...

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