On set R2 define ∼ by writing(a,b)∼(u,v)⇔ 2a−b =
2u−v. Prove that∼is an equivalence relation on...
On set R2 define ∼ by writing(a,b)∼(u,v)⇔ 2a−b =
2u−v. Prove that∼is an equivalence relation on R2
In the previous problem:
(1) Describe [(1,1)]∼. (That is formulate a statement P(x,y)
such that [(1,1)]∼ = {(x,y) ∈ R2 | P(x,y)}.)
(2) Describe [(a, b)]∼ for any given point (a, b).
(3) Plot sets [(1,1)]∼ and [(0,0)]∼ in R2.
Let S ∈ L(R2) be given by S(x1, x2) = (x1 +x2, x2) and let I...
Let S ∈ L(R2) be given by S(x1, x2) = (x1 +x2, x2) and let I ∈
L(R2) be the identity operator.
Using the inner product defined in problem 1 for the standard basis
and the dot product,
compute <S, I>, || S ||, and || I ||
{Inner product in problem 1: Let W be an inner product space and
v1, . . . , vn a basis of V. Show that <S, T> = <Sv1, T
v1> +...
Let R1(t) = < t2+3 , 2t +1, -t+3
>
Let R2(s) = < 2s ,...
Let R1(t) = < t2+3 , 2t +1, -t+3
>
Let R2(s) = < 2s , s+1 , s2+2s-6
>
Show that these two curves intersect at a right angle.
Let Q be the set {(a, b) ∶ a ∈ Z and b ∈ N}.
Let...
Let Q be the set {(a, b) ∶ a ∈ Z and b ∈ N}.
Let (a, b), (c, d) ∈ Q. Show that (a, b) ∼ (c, d) if and only if
ad − bc = 0 defines an equivalence relation on Q.
Problem 13.5. Consider a “square” S = {(x, y) : x, y ∈ {−2, −1,
0,...
Problem 13.5. Consider a “square” S = {(x, y) : x, y ∈ {−2, −1,
0, 1, 2}}. (a) Let (x, y) ∼ (x 0 , y0 ) iff |x| + |y| = |x 0 | + |y
0 |. It is an equivalence relation on S. (You don’t need to prove
it.) Write the elements of S/ ∼. (b) Let (x, y) ∼ (x 0 , y0 ) iff •
x and x 0 have the same sign (both...
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?
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...