Question

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

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 )

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 )

Let a be prime and b be a positive integer. Prove/disprove, that
if a divides b^2 then a divides b.

Let Z be the integers.
(a) Let C1 = {(a, a) | a ∈ Z}. Prove that
C1 is a subgroup of Z × Z.
(b) Let n ≥ 2 be an integer, and let Cn = {(a, b) | a
≡ b( mod n)}. Prove that Cn is a subgroup of Z × Z.
(c) Prove that every proper subgroup of Z × Z that contains
C1 has the form Cn for some positive integer
n.

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 G=Z x Z and H={ (a, b) in Z x Z | 8 divides (a+b) }.
1. Prove that G/H is isomorphic to Z8.
2. What is the index of [G : H]? Explain.

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.)

1)Let ? be an integer. Prove that ?^2 is even if and only if ?
is even. (hint: to prove that ?⇔? is true, you may instead prove ?:
?⇒? and ?: ? ⇒ ? are true.)
2) Determine the truth value for each of the following
statements where x and y are integers. State why it is true or
false. ∃x ∀y x+y is odd.

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...

3.a) Let n be an integer. Prove that if n is odd, then
(n^2) is also odd.
3.b) Let x and y be integers. Prove that if x is even and y is
divisible by 3, then the product xy is divisible by 6.
3.c) Let a and b be real numbers. Prove that if 0 < b < a,
then (a^2) − ab > 0.

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