Let X, Y ⊂ Z and x, y ∈ Z Let A = (X\{x}) ∪ {x}.
a) Prove or disprove: A ⊆ X
b) Prove or disprove: X ⊆ A
c) Prove or disprove: P(X ∪ Y ) ⊆ P(X) ∪ P(Y ) ∪ P(X ∩ Y )
d) Prove or disprove: P(X) ∪ P(Y ) ∪ P(X ∩ Y ) ⊆ P(X ∪ Y )
Prove Proposition 6.10 (Let f : X → Y and g : Y → Z be one
to one and onto functions. Then g ◦ f : X → Z is one to one and
onto; and (g ◦ f)−1 = f−1 ◦ g−1
).
Mystery(y, z: positive integer)
1 x=0
2 while z > 0
3 if z mod 2 ==1 then
4
x = x + y
5 y = 2y
6 z =
floor(z/2)
//floor is the rounding down operation
7 return x
Simulate this algorithm for y=4 and z=7 and answer the following
questions:
(3 points) At the end of the first execution of the while loop,
x=_____, y=______ and z=_______.
(3 points) At the end of the second execution of...
The curried version of let f (x,y,z) = (x,(y,z)) is
let f (x,(y,z)) = (x,(y,z))
Just f (because f is already curried)
let f x y z = (x,(y,z))
let f x y z = x (y z)
Let F be a field.
(a) Prove that the polynomials a(x, y) = x^2 − y^2, b(x, y) =
2xy and c(x, y) = x^2 + y^2 in F[x, y] form a Pythagorean triple.
That is, a^2 + b^2 = c^2. Use this fact to explain how to generate
right triangles with integer side lengths.
(b) Prove that the polynomials a(x,y) = x^2 − y^2, b(x,y) = 2xy
− y^2 and c(x,y) = x^2 − xy + y2 in F[x,y]...
Let S be the cone z = 4 − ( x^2 + y^2)^(1/2) where z ≥ 0,
oriented with downward pointing unit normal vectors. The image
below contains S. The grey plane is the xy-plane. Let F~ (x, y, z)
= <− z , x − y , x + y >. Use Stokes’ Theorem to evaluate Z Z
S ∇ × F~ · dS~.
let p = 1031, Find the number of solutions to the equation x^2
-2 y^2=1 (mod p), i.e., the number of elements (x,y),
x,y=0,1,...,p-1, which satisfy x^2 - 2 y^2=1 (mod p)