Consider the relation R defined on the real line R, and defined
as follows: x ∼...
Consider the relation R defined on the real line R, and defined
as follows: x ∼ y if and only if the distance from the point x to
the point y is less than 3. Study if this relation is reflexive,
symmetric, and transitive. Which points are related to 2?
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x,...
Consider the relation R defined on the set R as follows: ∀x, y ∈
R, (x, y) ∈ R if and only if x + 2 > y.
For example, (4, 3) is in R because 4 + 2 = 6, which is greater
than 3.
(a) Is the relation reflexive? Prove or disprove.
(b) Is the relation symmetric? Prove or disprove.
(c) Is the relation transitive? Prove or disprove.
(d) Is it an equivalence relation? Explain.
Let R be the relation on Z defined by:
For any a, b ∈ Z ,...
Let R be the relation on Z defined by:
For any a, b ∈ Z , aRb if and only if 4 | (a + 3b). (a) Prove that
R is an equivalence relation.
(b) Prove that for all integers a and b, aRb if and only if a ≡
b (mod 4)
13. Let R be a relation on Z × Z be defined as (a, b) R...
13. Let R be a relation on Z × Z be defined as (a, b) R (c, d)
if and only if a + d = b + c.
a. Prove that R is an equivalence relation on Z × Z.
b. Determine [(2, 3)].
4. Let A={(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}. The relation R
is defined on A as follows: For all (a, b),(c,...
4. Let A={(1,3),(2,4),(-4,-8),(3,9),(1,5),(3,6)}. The relation R
is defined on A as follows: For all (a, b),(c, d) ∈ A, (a, b) R (c,
d) ⇔ ad = bc . R is an equivalence relation. Find the distinct
equivalence classes of R.
Let A = R x R, and let a relation S be defined as: “(x1 ,...
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