Math Answers

1. State whether the following statements are True or False and also give the reason in support of your answer.

(a)Standard deviation of a random variable X may take any real value, i.e. its value lies in the interval (-infinity, infinity)

(b)If events E1,E2,...En are mutually exclusive and exhaustive then P(E1UE2U....UEn)will be greater than 1/2 but less than 1.

(c)If X is a random variable having range set {0, 1, 2, 3} then the set {0,1,2,3} then the set {x belongs to S:X(x)=0}is an event having at least one outcome of the random experiment

What is the truth value of each of the following formulas where the domain consists

of the integers? Justify your answers.

(a) ("\\forall"x)("\\forall"y)(x < y "\\lor" y < x)

(b) ("\\exists"9y)("\\forall"x)(x + y = 0)

There are 4 black, 3 blue and 8 red balls in an urn. Three balls are selected one by one without replacement. What is the probability that:

(i)First ball drawn is black, second one is red and third one is blue

(ii)All the three balls are of the same colour

7. Express these system specifications using the propositions

p “The message is scanned for viruses”

and q “The message was sent from an unknown system” together with logical connectives (including negations).

a) “The message is scanned for viruses whenever the message was sent from an unknown system.”

b) “The message was sent from an unknown system but it was not scanned for viruses.”

c) “It is necessary to scan the message for viruses whenever it was sent from an unknown system.”

d) “When a message is not sent from an unknown system it is not scanned for viruses.”

Suppose that a fair six-sided die is tossed. what is the probability of obtaining a number less or equal to three if even number have occured

The height of grade 1 pupils are approximately normally distributed with µ = 45 inches and s = 2.

1. If an individual pupil is selected at random, what is the probability that he or she has a height of 42 and 47?
2. A class of 30 of these pupils is used as a sample. What is the probability that the class mean is between 42 and 47?
3. If a pupil is selected at random, what is the probability that is taller than 46 inches?
4. A class of 30 of these pupils is used as sample. What is the probability that the class mean is greater than 46 inches?

Consider the following relations on {1, 2, 3, 4}.

R₁ = {(2,2), (2,3),(2,4),(3,2),(3,3),(3,4)}

R₂ = {(1,1),(1,2),(2,1),(2,2),(3,3),(4,4)}

R₃ = {2,4),(4,2)}

R₄ = {(1,2),(2,3),(3,4)}

R₅ = {(1,1),(2,2),(3,3),(4,4)}

a)   Which of these relations are reflexive? Justify your answers.

b)   Which of these relations are symmetric? Justify your answers.

c)   Which of these relations are antisymmetric? Justify your answer.

d)   Which of these relations are transitive? Justify your answers.

Let R₁ and R₂ be the relations on {1,2,3,4} given by

R₁ = {(1,1), (1,2), (3,4), (4,2)}

R₂ = {(1,1), (2,1), (3,1), (4,4), (2,2)}

List the elements of R_{1 °} R₂ and R_{2 °} R₁

Solve the PDE whose auxiliary equations as follows: 𝑑𝑥/ 2𝑦(𝑧 − 3) = 𝑑𝑦 /𝑦(2𝑥 − 𝑧) = 𝑑𝑧 /𝑦(2𝑥 − 3)

1.  Let A = {c, n, b}, B = {x, y} and C = {0, 1}.

Find

a)   A X B X C

b)   C X B X A

c)   B X C X C