In: Advanced Math

Suppose x,y ∈ R and assume that x < y. Show that for all z
∈ (x,y), there exists α ∈ (0,1) so that αx+(1−α)y = z. Now, also
prove that a set X ⊆ R is convex if and only if the set X satisfies
the property that for all x,y ∈ X, with x < y, for all z ∈
(x,y), z ∈ X.

Show that every Pythagorean triple (x, y, z) with x, y, z having
no common factor d > 1 is of the form (r^2 - s^2, 2rs, r^2 +
s^2) for positive integers r > s having no common factor > 1;
that is
x = r^2 - s^2, y = 2rs, z = r^2 + s^2.

Suppose X,Y,Z ⊆ U. If X is the set of all people who played
hockey in high school, Y is the set of all out-of-state students,
and Z is the set of all international students, describe the
following sets in words:
(a) X′ ∪Y ∪Z (b) X ∩ Y ′ ∩ Z′ (c)
(X∩Y′)∪Z′
Consider the set A = {b, c, d}.
(a) How many subsets does A have? (b) List all subsets of A.
Suppose that a committee of...

R is included in (R-{0} )x(R-{0} )
R = {(x,y) : xy >0}
Show that R is an equivalent relation and find f its equivalent
classes

Show that if (x,y,z) is a primitive Pythagorean triple, then X and
Y cannot both be even and cannot both be odd. Hint: for the odd
case, assume that there exists a primitive Pythagorean triple with
X and Y both odd. Then use the proposition "A perfect square always
leaves a remainder r=0 or r=1 when divided by 4." to produce a
contradiction.

Given the function u(p,q,r)=((p-q)/(q-r)), with p=x+y+z,q=x-y+z,
and r=x+y-z, find the partial derivatives au/ax=, au/ay=,
au/az=

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)

Find the positive numbers x, y and z whose sum is 100 such that
x^r y^s z^t is a maximum, where r, s and t are constants.

6. Let R be a relation on Z x Z such that for all ordered pairs
(a, b),(c, d) ∈ Z x Z, (a, b) R (c, d) ⇔ a ≤ c and b|d . Prove that
R is a partial order relation.

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 )

Show that {xx^R | x,y ∈ {0,1}*} is a context-free language.
Note that x^R is the reversal of x.
Show all work.
Question is for Discrete Math Structures

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