Question

Determine if each of the following statements is true or false. If a statement is true, then write a formal proof of that statement, and if it is false, then provide a counterexample that shows its false.

1) For each integer *a* there exists an integer
*n* such that *a* divides (8*n* +7) and
*a* divides (4n+1), then a divides 5.

2)For each integer *n* if *n* is odd, then 8
divides (*n*^{4}+4*n*^{2}+11).

Answer #1

Write a formal proof to prove the following conjecture to be
true or false.
If the statement is true, write a formal proof of it. If the
statement is false, provide a counterexample and a slightly
modified statement that is true and write a formal proof of your
new statement.
Conjecture: There does not exist a pair of integers m and n such
that m^2 - 4n = 2.

Determine if the following statements are true or false.
In either case, provide a formal proof using the definitions of
the big-O, big-Omega, and big-Theta notations. For instance, to
formally prove that f (n) ∈ O(g(n)) or f (n) ∉ O(g(n)), we need to
demonstrate the existence of a constant c and a sufficient large n0
such that f (n) ≤ c g(n) for all n ≥ n0, or showing that there are
no such values.
a) [1 mark] 10000n2...

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

For each of the following statements: if the statement is true,
then give a proof; if the
statement is false, then write out the negation and prove that.
For all sets A;B and C, if B n A = C n A, then B = C.

Determine whether the following statements are true or
false. If the statement is false, then explain why the statement is
false or rewrite the statement so that it is true.
If a scatterplot shows a linear association
between two numerical variables, then a correlation coefficient
that is close to 1 indicates a strong positive trend and a
correlation coefficient that is close to 0 indicates a strong
negative trend.

1) (a) Determine if the following statements are true or false.
If true give a reason or cite a theorem and if false, give a
counterexample.
i) If { a n } is bounded, then it converges.
ii) If { a n } is not bounded, then it diverges.
iii) If { a n } diverges, then it is not bounded.
(b) Give an example of divergent sequences { a n } and
{ b n } such that {...

3. For each of the following statements, either provide a short
proof that it is true (or appeal to the deﬁnition) or provide a
counterexample showing that it is false.
(e) Any set containing the zero vector is linearly
dependent.
(f) Subsets of linearly dependent sets are linearly
dependent.
(g) Subsets of linearly independent sets are linearly
independent.
(h) The rank of a matrix is equal to the number of its nonzero
columns.

Determine whether the given statement is true or false. Explain
your answer.
(a) If R is an antisymmetric relation, then R is not
symmetric.
(b) If John Jay College was founded in 1997, then the moon is
made of cheese. (
c) ∀x∃y(x divides y) where the domain of discourse for both
variables is {2, 3, 4, 5, 6}.
(d) ∃x∀y(x divides y) where the domain of discourse for both
variables is {2, 3, 4, 5, 6}.
(e) ∀n(3n ≤...

For each of the following true/false statements. Indicate if the
statement is true or false. If false, correct the statement to be
true.
A. _____ Fas ligand binding with Fas receptor stimulates a
specific G protein
B. _____ A mutation in Bax that blocks Bax binding to the
mitochondria will promote apoptosis
C. _____ Telomerase activity is off in most somatic cells in
adults
D. _____ The telomerase enzyme extends the telomeres using its
RNA polymerase activity.

True Or False
1. If nn is odd and the square root of nn is a natural number
then the square root of nn is odd.
2. The square of any even integer is even
3. The substraction of 2 rational numbers is rational.
4. If nn is an odd integer, then n2+nn2+n is even.
5. If a divides b and a divides c then a divides bc.
6. For all real numbers a and b, if a^3=b^3 then a=b.

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