Question

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

Answer #1

Prove or disprove the following statements.
a) ∀a, b ∈ N, if ∃x, y ∈ Z and ∃k ∈ N such that ax + by = k,
then gcd(a, b) = k
b) ∀a, b ∈ Z, if 3 | (a 2 + b 2 ), then 3 | a and 3 | b.

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 )

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 or disprove that there do not exist z, y,
and z are positive integers such that X7 - Y5
= Z4

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

When we say Prove or disprove the
following statements, “Prove” means you show the
statement is true proving the correct statement using at most 3
lines or referring to a textbook theorem.
“Disprove” means you show a statement is wrong by
giving a counterexample why that is not true).
Are the following statements true or not? Prove or disprove
these one by one. Show how the random variable X looks in each
case.
(a) E[X] < 0 for some random...

5. Prove or disprove the following statements:
(a) Let R be a relation on the set Z of integers such that xRy
if and only if xy ≥ 1. Then, R is irreflexive.
(b) Let R be a relation on the set Z of integers such that xRy
if and only if x = y + 1 or x = y − 1. Then, R is irreflexive.
(c) Let R and S be reflexive relations on a set A. Then,...

. Let f : Z → N be function.
a. Prove or disprove: f is not strictly increasing. b. Prove
or disprove: f is not strictly decreasing.

Prove or disprove (a) Z[x]/(x^2 + 1), (b) Z[x]/(x^2 - 1) is an
Integral domain.
By showing (a) x^2+1 is a prime ideal or showing x^2 + 1 is not
prime ideal.
By showing (b) x^2-1 is a prime ideal or showing x^2 - 1 is not
prime ideal.
(Hint: R/I is an integral domain if and only if I is a prime
ideal.)

Prove or disprove the following statements. Remember to disprove
a statement you have to show that the statement is false.
Equivalently, you can prove that the negation of the statement is
true. Clearly state it, if a statement is True or False. In your
proof, you can use ”obvious facts” and simple theorems that we have
proved previously in lecture.
(a) For all real numbers x and y, “if x and y are irrational,
then x+y is irrational”.
(b) For...

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