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.
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...
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.
Recall from class that we defined the set of integers by
defining the equivalence relation ∼...
Recall from class that we defined the set of integers by
defining the equivalence relation ∼ on N × N by (a, b) ∼ (c, d) =⇒
a + d = c + b, and then took the integers to be equivalence classes
for this relation, i.e. Z = [(a, b)]∼ | (a, b) ∈ N × N . We then
proceeded to define 0Z = [(0, 0)]∼, 1Z = [(1, 0)]∼, − [(a, b)]∼ =
[(b, a)]∼, [(a, b)]∼...
1. We define a relation C on the set of humans as xRy ⇐⇒ x and...
1. We define a relation C on the set of humans as xRy ⇐⇒ x and y
were born in the same country
Describe the equivalence class containing yourself as an
element.
2. Let R be an equivalence relation with (x, y) ∈ R and (y, z)
is not ∈ R (that is, y does not relate to z). Can you determine
whether or not xRz? Why or why not?
Consider the following set S = {(a,b)|a,b ∈ Z,b 6= 0} where Z
denotes the integers....
Consider the following set S = {(a,b)|a,b ∈ Z,b 6= 0} where Z
denotes the integers. Show that the relation (a,b)R(c,d) ↔ ad = bc
on S is an equivalence relation. Give the equivalence class
[(1,2)]. What can an equivalence class be associated with?