Question

3. Prove or disprove: For integers a and b, if a|b, then a^2|b^2.

4. Suppose that for sets A,B,C, and D,A∩B⊆C∩D and A⊆C\D. Prove that A and B are disjoint.

Answer #1

8. Let a, b be integers. (a) Prove or disprove: a|b ⇒ a ≤ b. (b)
Find a condition on a and/or b such that a|b ⇒ a ≤ b. Prove your
assertion! (c) Prove that if a, b are not both zero, and c is a
common divisor of a, b, then c ≤ gcd(a, b).

Prove or disprove each of the following statements:
(a) For all integers a, a | 0.
(b) For all integers a, 0 | a.
(c) For all integers a, b, c, n, and m, if a | b and a | c, then
a | (bn+cm).

3. Prove or disprove the following statement: If A and B are
finite sets, then |A ∪ B| = |A| + |B|.

Suppose that A, B and C are events. Prove or disprove the
statement “A, B and C are mutually exclusive if and only if A,
B and C are exhaustive”.

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

Prove or Disprove
Suppose we construct arrays of integers. Let S be the
set of all arrays which are arranged in sorted order. The set S is
decidble.
A Turing machine with two tapes is no more powerful
than a Turing machine with one tape. (That is, both types of
machines can compute the same set of functions.)

Let us say that two integers are near to one another provided
their difference is 2 or smaller (i.e., the numbers are at most 2
apart). For example, 3 is near to 5, 10 is near to 9, but 4 is not
near to 8. Let R stand for this is-near-to relation. (a) Write down
R as a set of ordered pairs. Your answer should look like this: R =
{(x, y) : . . .}. (b) Prove or disprove:...

Let A, B, C and D be sets. Prove that A \ B and C \ D are
disjoint if and only if A ∩ C ⊆ B ∪ D.

Prove by contradiction: Let a and b be integers. Show that if is
odd, then a is odd and b is odd.
a) State the negation of the above implication.
b) Disprove the negation and complete your proof.

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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