3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine...
3. For each of the piecewise-defined functions f, (i) determine
whether f is 1-1; (ii) determine whether f is onto. Prove your
answers.
(a) f : R → R by f(x) = x^2 if x ≥ 0, 2x if x < 0.
(b) f : Z → Z by f(n) = n + 1 if n is even, 2n if n is odd.
Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define
d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|.
Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…).
Prove that maps X onto...
Let X=2N={x=(x1,x2,…):xi∈{0,1}} and define
d(x,y)=2∑(i≥1)(3^−i)*|xi−yi|.
Define f:X→[0,1] by f(x)=d(0,x), where 0=(0,0,0,…).
Prove that maps X onto the Cantor set and satisfies
(1/3)*d(x,y)≤|f(x)−f(y)|≤d(x,y) for x,y∈2N.
Determine which of the following functions are injective
(one-to-one) on their respective domains and codomains
(a)...
Determine which of the following functions are injective
(one-to-one) on their respective domains and codomains
(a) f : ℝ → [0,∞), where f(x) = x²
(b) g : ℕ → ℕ, where g(x) = 3x − 2
(c) h : ℤ_7 → ℤ_7, where h(x) ≡ 5x + 2 (mod 7)
(d) p : ℕ ⋃ {0} → ℕ ⋃ {0}, where p(x) = x div 3
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...
The function f(x, y) is defined by
f(x, y) = 5x^3 * cos(y^3).
You will compute...
The function f(x, y) is defined by
f(x, y) = 5x^3 * cos(y^3).
You will compute the volume of the 3D body below z = f(x, y) and
above the x, y-plane, when x
and y are bounded by the region defined between y = 2 and y =1/4 *
x^2.
(a) First explain which integration order is the preferred one in
this case and explain why.
(b) Then compute the volume.