Question

(a) Let the statement,

∀x∈R,∃y∈R G(x,y), be true for predicate G(x,y).

For each of the following statements, decide if the statement is certainly true, certainly false,or possibly true, and justify your solution.

1

(i)

G(3,4)

(ii)

∀x∈RG(x,3)

(iii)

∃y G(3,y)

(iv)

∀y¬G(3,y)(v)∃x G(x,4)

Answer #1

In each case below show that the statement is True or give an
example showing that it is False.
(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is
independent.
(ii) If {X, Y, Z} is independent in R n, then {Y, Z} is
independent.
(iii) If {Y, Z} is dependent in R n, then {X, Y, Z} is
dependent.
(iv) If A is a 5 × 8 matrix with rank A...

Prove: Let x,y be in R such that x < y.
There exists a z in R such that x < z <
y.
Given:
Axiom 8.1. For all x,y,z in
R:
(i) x + y = y + x
(ii) (x + y) + z = x + (y + z)
(iii) x*(y + z) = x*y + x*z
(iv) x*y = y*x
(v) (x*y)*z = x*(y*z)
Axiom 8.2. There exists a real number 0 such that
for all...

Let P be a predicate. Determine whether or not each of the
following implications is true and give a brief English explanation
for your answer.
1)∀x∃yP(x, y) -> ∃y∀xP(x, y)
2)∃y∀xP(x, y) -> ∀x∃yP(x, y)

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) LetX andY besetsfromRn. IfX⊂Y thenX is closed if and only if
Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then eitherX or Y is closed and convex (one or the other).
(iii) LetX beanopensetandY ⊆X. IfY ≠∅,thenY isaconvexset.
(iv) SupposeX isanopensetandY isaconvexset. IfX∩Y ⊂X then
X∪Y...

Exercise 4.11. For each of the following, state whether it is
true or false. If true, prove. If false, provide a
counterexample.
(i) Let X and Y be sets from Rn. If X ⊂ Y then X is closed if
and only if Y is closed.
(ii) Let X and Y be sets from Rn. If X ∩Y is closed and convex
then either X or Y is closed and convex (one or the other).
(iii) Let X be an...

Let f : R → R be defined by f(x) = x^3 + 3x, for all x. (i)
Prove that if y > 0, then there is a solution x to the equation
f(x) = y, for some x > 0. Conclude that f(R) = R. (ii) Prove
that the function f : R → R is strictly monotone. (iii) By
(i)–(ii), denote the inverse function (f ^−1)' : R → R. Explain why
the derivative of the inverse function,...

4. Let A = {1, 2, 3, 4, 5}. Let L = {(x, y) ∈ A × A : x < y}
and B = {(x, y) ∈ A × A : |x − y| = 1}.
i. Draw graphs representing L and B.
ii. Determine L ◦ B.
iii. Determine B ◦ B.
iv. Is B transitive? Explain.

[2 marks] Find the derivative of y =
√
4 sin x + 6
at x = 0.
Consider the following statements. The limit
lim
x→0
g(4 + h) − g(4)
h
is equivalent to:
(i) The derivative of g(x) at x =
h
(ii) The derivative of g(x + 4) at x
= 0
(iii) The derivative of g(−x) at x
= −4
Determine which of the above statements are True (1) or False
(2).
If f (3) = ...

Let u, v, and w be vectors in Rn. Determine which of the
following statements are always true. (i) If ||u|| = 4, ||v|| = 5,
and ?||u + v|| = 8, then u?·?v = 4. (ii) If ||u|| = 2 and ||v|| =
3, ?then |u?·?v| ? 5. (iii) The expression (v?·?w)u is both
meaningful and defined. (A) (ii) and (iii) only (B) (ii) only (C)
none of them (D) all of them (E) (i) only (F) (i) and...

For each of the following statements, identify whether the
statement is true or false, and explain why. Please limit each
response to no more than 3 sentences.
i) A p-value is the probability that the null hypothesis is
false. ii) A chi-square test statistic can never be negative.
iii) If we reject the null hypothesis that a population
proportion is equal to a specific value, then that specific value
will not be contained in the associated confidence interval.
iv) If...

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