2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff...
2. Define a relation R on pairs of real numbers as follows: (a,
b)R(c, d) iff either a < c or both a = c and b ≤ d. Is R a
partial order? Why or why not? If R is a partial order, draw a
diagram of some of its elements.
3. Define a relation R on integers as follows: mRn iff m + n is
even. Is R a partial order? Why or why not? If R is...
Given a relation R(A, B, C, D, E) with the following FD
Set
FD = {...
Given a relation R(A, B, C, D, E) with the following FD
Set
FD = { A→C, B→C, C→D, DE→A, CE→A}
Suppose we decompose it into R1(A, D), R2(A, B), R3(B, E), R4(C, D,
E) and R5(A, E), is it a lossless decomposition? Show your
proof.
Let R be the relation on the set of real numbers such that xRy
if and...
Let R be the relation on the set of real numbers such that xRy
if and only if x and y are real numbers that differ by less than 1,
that is, |x − y| < 1. Which of the following pair or pairs can
be used as a counterexample to show this relation is not an
equivalence relation?
A) (1, 1)
B) (1, 1.8), (1.8, 3)
C) (1, 1), (3, 3)
D) (1, 1), (1, 1.5)
Let A=NxN and define a relation on A by (a,b)R(c,d) when a⋅b=c⋅d
a ⋅ b =...
Let A=NxN and define a relation on A by (a,b)R(c,d) when a⋅b=c⋅d
a ⋅ b = c ⋅ d . For example, (2,6)R(4,3)
a) Show that R is an equivalence relation.
b) Find an equivalence class with exactly one element.
c) Prove that for every n ≥ 2 there is an equivalence class with
exactly n elements.
A relation R on a set A is called circular if for all a,b,c in
A,...
A relation R on a set A is called circular if for all a,b,c in
A, aRb and bRc imply cRa. Prove that a relation is an equivalence
relation iff it is reflexive and circular.
Define a relation on N x N by (a, b)R(c, d) iff ad=bc
a. Show that...
Define a relation on N x N by (a, b)R(c, d) iff ad=bc
a. Show that R is an equivalence relation.
b. Find the equivalence class E(1, 2)
Determine whether the given relation is an equivalence relation
on {1,2,3,4,5}. If the relation is an...
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
List all the ordered pairs in the relation
R = {(a, b) | b divides a}...
List all the ordered pairs in the relation
R = {(a, b) | b divides a} on the set {1, 2, 3, 4, 5, 6}
Let P be the set of all ordered pairs (a, b) where a and b are...
Let P be the set of all ordered pairs (a, b) where a and b are
real numbers. Let us define a two-place relation ≡ on P by (a, b) ≡
(c, d) if and only if a^2 − c^2 = 2b − 2d where (a, b) and (c, d)
belong to P. Prove that ≡ is an equivalence relation on P. Draw a
diagram on the X × Y plane of the equivalence class that contains
the point (2,...