Question

Determine whether the relations R1 and R2 are equivalence relations to the specified quantity and, if necessary, determine the corresponding equivalence classes.

∀x, y ∈Z: x ~R1 y ⇐⇒ x + y is divisible by 2,

∀ g, h ∈ {t: t is a straight line in R2}: g ~R2 h ⇐⇒ g and h have common Points.

Answer #1

Determine whether the relations R1 and R2 are equivalence
relations to the specified quantity and, if necessary, determine
the corresponding equivalence classes. ∀x, y ∈Z: x ~R1 y ⇐⇒ x + y
is divisible by 2, ∀ g, h ∈ {t: t is a straight line in R2}: g ~R2
h ⇐⇒ g and h have common Points.

Let R1 and R2 be equivalence relations on a set A. (a) Must
R1∪R2 be an equivalence relation? (b) Must R1∩R2 be an equivalence
relation? (c) Must R1⊕R2 be an equivalence relation?[⊕is the
symmetric difference:x∈A⊕B if and only if x∈A,x∈B, and x
/∈A∩B.]

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),...

Two spacecraft are following paths in space given by
r1=〈sin(t),t,t^2〉r1=〈sin(t),t,t^2〉 and
r2=〈cos(t),1−t,t^3〉.r2=〈cos(t),1−t,t^3〉. If the temperature for
the points is given by T(x,y,z)=x^2y(5−z),T(x,y,z)=x^2y(5−z), use
the Chain Rule to determine the rate of change of the difference D
in the temperatures the two spacecraft experience at time t=2.
(Use decimal notation. Give your answer to two decimal
places.)

Determine whether or not the following relations have the
properties of the relations. Be sure to justify your answers.
a) R = {(x, y)| y is a biological parent of x} on the set of all
people
b) R = {(x, y) ∈ N × N | lcm(x, y) = 10}

Two spacecraft are following paths in space given by
r1=〈sin(t),t,t^2〉 and r2=〈cos(t),1−t,t^3〉. If the temperature for
the points is given by T(x,y,z)=x^2y(1−z), use the Chain Rule to
determine the rate of change of the difference D in the
temperatures the two spacecraft experience at time t=3.
(Use decimal notation. Give your answer to two decimal
places.)

Two spacecraft are following paths in space given by
r1=〈sin(t),t,t^2〉 and r2=〈cos(t),1−t,t^3〉. If the temperature for
the points is given by T(x,y,z)=x^2y(8−z), use the Chain Rule to
determine the rate of change of the difference D in the
temperatures the two spacecraft experience at time t=1.
(Use decimal notation. Give your answer to two decimal
places.)

Determine whether the given relation is an equivalence relation
on {1,2,3,4,5}. If the relation is an equivalence relation, list
the equivalence classes (x, y E {1, 2, 3, 4, 5}.)
{(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1), (3,4),
(4,3)}
If the relation above is not an equivalence relation, state that
the relation is not an equivalence relation and why.
Example: "Not an equivalence relation. Relation is not
symmetric"
Remember to test all pairs in relation R

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}.

Relations and Functions
Usual symbols for the above are;
Relations: R1, R2, S, T, etc
Functions: f, g, h, etc. But remember a function is a special
kind of relation so it might turn out that a Relation, R, is a
function, too.
Relations
To understand the symbolism better, let’s say the domain of a
relation, R, is A = { a, b , c} and the Codomain is B = {
1,2,3,4}.
Here is the relation: a R 1, ...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 7 minutes ago

asked 21 minutes ago

asked 31 minutes ago

asked 33 minutes ago

asked 45 minutes ago

asked 51 minutes ago

asked 52 minutes ago

asked 59 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago