Let R be a relation on a set that is reflexive and symmetric but
not transitive? Let R(x) = {y : x R y}. [Note that R(x) is the same
as x / R except that R is not an equivalence relation in this
case.] Does the set A = {R(x) : x ∈ A} always/sometimes/never form
a partition of A? Prove that your answer is correct. Do not prove
by examples.
Question: Consider the relation R on A defined by aRb iff 1mod4
= bmod4
a)Construct the diagraph for this relation
b)show that R is an equivalence relation
Part B: Now consider the relation R on A defined by aRb iff a
divides b (Divides relation)
c) Show that R is partial ordering
d) Contruct the hasse diagram for this relation
let
G be a simple graph. show that the relation R on the set of
vertices of G such that URV if and only if there is an edge
associated with (u,v) is a symmetric irreflexive relation on
G
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.
Determine whether the relation R on the set of all
people is reflexive, symmetric, antisymmetric, and/or transitive,
where (a, b) ∈ R if and only if
a) a is taller than b.
b) a and b were born on the same day.
Let S = {1,2,3,4} and let A = SxS
Define a relation R on A by (a,b)R(c,d) iff ad = bc
Write out each equivalence class (by "write out" I mean tell me
explicitly which elements of A are in each equivalence class)
Hint: |A| = 16 and there are 11 equivalence classes, so there
are several equivalence classes that consist of a single element of
A.
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.
Let
A = Σ*, and let R be the relation "shorter than." Determine whether
or not the given relation R, on the set A, is reflexive, symmetric,
antisymmetric, or transitive.