Question

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 4

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 )

Answer #1

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 )

If X and Y are correlated and Y and Z are correlated, then X and
Z are correlated.
prove or disprove?

Prove or disprove following by giving examples:
(a) If X ⊂ Y and X ⊂ Z, then X ⊂ Y ∩ Z
(b) If X ⊆ Y and Y ⊆ Z, then X ⊆ Z
(c) If X ∈ Y and Y ∈ Z, then X ∈ Z

Let F = {A ⊆ Z : |A| < ∞} be the set of all finite sets of
integers. Let R be the relation on F defined by A R B if and only
if |A| = |B|. (a) Prove or disprove: R is reflexive. (b) Prove or
disprove: R is irreflexive. (c) Prove or disprove: R is symmetric.
(d) Prove or disprove: R is antisymmetric. (e) Prove or disprove: R
is transitive. (f) Is R an equivalence relation? Is...

Let A = {x ∈ Z | x = 5a+2 for some integer a}, B = {x ∈ Z | x =
10b−3 for some integer b}. Prove or disprove the statements. 1. A ⊆
B 2. B ⊆ A

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is surjective then g is surjective.
8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
composition g o f is bijective then f is bijective.
8.6: Let f : X → Y and g : Y→ Z be maps. Prove that if
composition g o f is bijective then f is...

Using field and order axioms prove the following theorems:
(i) Let x, y, and z be elements of R, the
a. If 0 < x, and y < z, then xy < xz
b. If x < 0 and y < z, then xz < xy
(ii) If x, y are elements of R and 0 < x < y, then 0 <
y ^ -1 < x ^ -1
(iii) If x,y are elements of R and x <...

let G be a group of order 18. x, y, and z are elements
of G. if | < x, y >| = 9 and o(z) = order of z = 9, prove
that G = < x, y, z >

Let
x = {x} and y ={y} represent bounded sequences of real numbers, z =
x + y, prove the following: supX + supY = supZ where sup represents
the supremum of each sequence.

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

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