Consider the following relation on the set Z: xRy ?
x2 + y is even.
For...
Consider the following relation on the set Z: xRy ?
x2 + y is even.
For each question below, if your answer is "yes", then prove it, if
your answer is "no", then show a counterexample.
(i) Is R reflexive?
(ii) Is R symmetric?
(iii) Is R antisymmetric?
(iv) Is R transitive?
(v) Is R an equivalence relation? If it is, then describe the
equivalence classes of R. How many equivalence classes are
there?
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.
Determine whether the binary relation R on {a, b,
c} where R={(a, a), (b, b)), (c,...
Determine whether the binary relation R on {a, b,
c} where R={(a, a), (b, b)), (c, c), (a, b), (a,
c), (c, b) } is:
a.
reflexive, antisymmetric, symmetric
b.
transitive, symmetric, antisymmetric
c.
antisymmetric, reflexive, transitive
d.
symmetric, reflexive, transitive
Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let...
Let Let A = {a, e, g} and B = {c, d, e, f, g}. Let f : A → B and
g : B → A be defined as follows: f = {(a, c), (e, e), (g, d)} g =
{(c, a), (d, e), (e, e), (f, a), (g, g)}
(a) Consider the composed function g ◦ f.
(i) What is the domain of g ◦ f? What is its codomain?
(ii) Find the function g ◦ f. (Find...
Construct a binary relation R on a nonempty set A satisfying the
given condition, justify your...
Construct a binary relation R on a nonempty set A satisfying the
given condition, justify your solution.
(a) R is an equivalence relation.
(b) R is transitive, but not symmetric.
(c) R is neither symmetric nor reflexive nor transitive.
(d) (5 points) R is antisymmetric and symmetric.
Let A = {1,2,3,4,5} and X = P(A) be its powerset. Define a
binary relation on...
Let A = {1,2,3,4,5} and X = P(A) be its powerset. Define a
binary relation on X by for any sets S, T ∈ X, S∼T if and only if S
⊆ T.
(a) Is this relation reflexive?
(b) Is this relation symmetric or antisymmetric?
(c) Is this relation transitive?
Let a, b, and n be integers with n > 1 and (a, n) = d....
Let a, b, and n be integers with n > 1 and (a, n) = d.
Then
(i)First prove that the equation a·x=b has solutions in n if and
only if d|b.
(ii) Next, prove that each of u, u+n′, u+ 2n′, . . . , u+
(d−1)n′ is a solution. Here,u is any particular solution guaranteed
by (i), and n′=n/d.
(iii) Show that the solutions listed above are distinct.
(iv) Let v be any solution. Prove that v=u+kn′ for...
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 S = {A, B, C, D, E, F, G, H, I, J} be the set...
Let S = {A, B, C, D, E, F, G, H, I, J} be the set consisting of
the following elements:
A = N, B = 2N , C = 2P(N) , D = [0, 1), E = ∅, F = Z × Z, G = {x
∈ N|x 2 + x < 2}, H = { 2 n 3 k |n, k ∈ N}, I = R \ Q, J =
R.
Consider the relation ∼ on S given...