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

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 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 )

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 =
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B 2. B ⊆ A

8.4: Let f : X → Y and g : Y→ Z be maps. Prove that if
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8.5: Let f : X → Y and g : Y→ Z be bijections. Prove that if
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Let X, Y and Z be sets. Let f : X → Y and g : Y → Z functions.
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Z and functions f : X → Y and g : Y → Z such that g ◦ f is
injective but g is not injective. (c) (3 Pts.) Show that...

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 x, y ∈Z. Prove that (x+1)y^2 is even if and only if x is odd
and y is even.

Let a,b ∈ Z. Prove that a−b is even if and only if x and y are
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