Question

Let D = E = {−2, 0, 2, 3}. Write negations for each of the following statements and determine which is true, the given statement or its negation. Explain your answer

(i) ∃x ∈ D such that ∀y ∈ E, x + y = y.

(ii) ∀x ∈ D, ∃y ∈ E such that xy ≥ y.

Answer #1

Let D = E = {−2, 0, 2, 3}. Write negations for each of the
following statements and determine which is true, the given
statement or its negation. Explain your answer.
(i) ∃x ∈ D such that ∀y ∈ E, x + y = y.
(ii) ∀x ∈ D, ∃y ∈ E such that xy ≥ y.

For each of the following statements:
• Rewrite the symbolic sentence in words,
• Determine if the statement is true or false and justify your
answer,• Negate the statement (you may write the negation
symbolically).
√
(a) ∀x∈R, x2 =x.
(b) ∃y∈R,∀x∈R,xy=0.
(c) ∃x ∈ R, ∀y ∈ R, x2 + y2 > 9.

(1) Determine whether the propositions p → (q ∨ ¬r) and (p ∧ ¬q)
→ ¬r are logically equivalent using either a truth table or laws of
logic.
(2) Let A, B and C be sets. If a is the proposition “x ∈ A”, b
is the proposition “x ∈ B” and
c is the proposition “x ∈ C”, write down a proposition involving a,
b and c that is logically equivalentto“x∈A∪(B−C)”.
(3) Consider the statement ∀x∃y¬P(x,y). Write down a...

2. Which of the following is a negation for ¡°All dogs are
loyal¡±? More than one answer may be correct.
a. All dogs are disloyal. b. No dogs are loyal.
c. Some dogs are disloyal. d. Some dogs are loyal.
e. There is a disloyal animal that is not a dog.
f. There is a dog that is disloyal.
g. No animals that are not dogs are loyal.
h. Some animals that are not dogs are loyal.
3. Write a...

(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)

Given a random variable X has the following pmf:
X
-1
0
1
P[X]
0.25
0.5
0.25
Define Y = X2 & W= Y+2.
Which one of the following statements is not true?
A) V[Y] = 0.25.
B) E[XY] = 0.
C) E[X3] = 0.
D) E[X+2] = 2.
E) E[Y+2] = 2.5.
F) E[W+2] = 4.5.
G) V[X+2] = 0.5.
H) V[W+2] = 0.25.
I) P[W=1] = 0.5
J) X and W are not independent.

[3 marks]
Consider the following statements about solutions
f (x) of the differential equation
y′ = (xy −
7y − 9x +
63)esin x.
(i) There is no k such that f (x)
= k is a solution.
(ii) If f (x) < 9 then
f (x) is decreasing for x <
0.
(iii) If f (x) < 0, then
f (x) is increasing when x >
7.
Determine which of the above statements are True or False .

Exercise 1.(50 pts) Translate the following sentences to
symbols, write a useful negation, and translate back to English the
negation obtained.
1. No right triangle is isoceles
2. For every positive real number x, there is a unique real
number y such that 2^y=x
.Exercise 2.(50 pts) Which of the following are true? Explain
your answer.
1.(∀x∈R)(x^2+ 6x+ 5 ≥ 0).
2.(∃x∈N)(x^2−x+ 41 is prime)

Suppose that X, Y, and Z are independent, with E[X]=E[Y]=E[Z]=2,
and E[X2]=E[Y2]=E[Z2]=5.
Find cov(XY,XZ).
(Enter a numerical answer.)
cov(XY,XZ)=
Let X be a standard normal random variable. Another random
variable is determined as follows. We flip a fair coin (independent
from X). In case of Heads, we let Y=X. In case of Tails, we let
Y=−X.
Is Y normal? Justify your answer.
yes
no
not enough information to determine
Compute Cov(X,Y).
Cov(X,Y)=
Are X and Y independent?
yes
no
not...

Let f ( x ) = cot x for 0 < x < π .
(a) State the range of f and graph it on the interval
given above.
(b) State the domain and range of g ( x ) = cot − 1 x . Graph
g ( x ) .
(c) State whether the graph increases or decreases on its
domain.
(d) Find each of the following limits based on the graph of g (
x...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 5 minutes ago

asked 15 minutes ago

asked 18 minutes ago

asked 30 minutes ago

asked 42 minutes ago

asked 48 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago