Question

- 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 ? = |?|.

Answer #1

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.

For each of the following relations, determine whether the
relation is reﬂexive, irreﬂexive, symmetric, antisymmetric, and/or
transitive. Then ﬁnd R−1.
a) R = {(x,y) : x,y ∈Z,x−y = 1}.
b) R = {(x,y) : x,y ∈N,x|y}.

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 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

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...

Determine whether the relation R is reflexive, symmetric,
antisymmetric, and/or transitive [4 Marks]
22
The relation R on Z where (?, ?) ∈ ? if ? = ? .
The relation R on the set of all subsets of {1, 2, 3, 4} where
SRT means S C T.

the relation R on the set of all people where aRb means that a
is younger than b. Determine if R is:
reflexive
symmetric
transitive
antisymmetric

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.

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