Search & Filtering
Let p and q be the following propositions:
p: Angela misses the final exam.
q : Angela passes the course.
Write the following propositions using p, q and negations and logical connectives:
(a) Angela either misses the final exam, or she does not pass the course, or both.
This question has 2 parts. Part 1: Suppose that F and X are events from a common sample space with P(F) 6= 0 and P(X) 6= 0. (a) Prove that P(X) = P(X|F)P(F) + P(X|F¯)P(F¯). Hint: Explain why P(X|F)P(F) = P(X ∩ F) is another way of writing the definition of conditional probability, and then use that with the logic from the proof of Theorem 4.1.1. (b) Explain why P(F|X) = P(X|F)P(F)/P(X) is another way of stating Theorem 4.2.1 Bayes Theorem. Part 2: A website reports that 70% of its users are from outside a certain country. Out of their users from outside the country, 60% of them log on every day. Out of their users from inside the country, 80% of them log on every day. (a) What percent of all users log on every day? Hint: Use the equation from Part 1 (a). (b) Using Bayes Theorem, out of users who log on every day, what is the probability that they are from inside the country?
USE THE METHOD OF DIRECT TO PROVE:
If a is an odd integer, then a’2 + 3a +5 is odd.
{F} Construct the Combinatorial Circuit of the given output.
- (p∨q) 2. (p∧ ~q)
- (p→~q)→(p∨q)
{F} 2.
Find the transitive closures of these relations on {1, 2, 3, 4}.
a) {(1, 2), (2,1), (2,3), (3,4), (4,1)}
b) {(2, 1), (2,3), (3,1), (3,4), (4,1), (4, 3)}
c) {(1, 2), (1,3), (1,4), (2,3), (2,4), (3, 4)}
d) {(1, 1), (1,4), (2,1), (2,3), (3,1), (3, 2), (3,4), (4, 2)}
3.
Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is
a) reflexive and transitive.
b) symmetric and transitive.
c) reflexive, symmetric, and transitive.
4.
Which of these relations on {0, 1, 2, 3} are equivalence relations? Determine the properties of an equivalence relation that the others lack.
a) {(0, 0), (1, 1), (2, 2), (3, 3)}
b) {(0, 0), (0, 2), (2, 0), (2, 2), (2, 3), (3, 2), (3, 3)}
c) {(0, 0), (1, 1), (1, 2), (2, 1), (2, 2), (3, 3)}
d) {(0, 0), (1, 1), (1, 3), (2, 2), (2, 3), (3, 1), (3, 2),(3, 3)}
e) {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (2, 0),(2, 2), (3, 3)}
{F} Construct a relation on the set {a, b, c, d} that is a. reflexive, symmetric, but not transitive. b. irreflexive, symmetric, and transitive. c. irreflexive, antisymmetric, and not transitive. d. reflexive, neither symmetric nor antisymmetric, and transitive. e. neither reflexive, irreflexive, symmetric, antisymmetric, nor transitive.
{F} How many ways can we get an even sum when two distinguishable dice are rolled ?
SHOW that every open interval is uncountable
When planning a party you want to know whom to invite. Among the people you would like to invite are three touchy friends. You know that if Jasmine attends, she will become unhappy if Samir is there, Samir will attend only if Kanti will be there, and Kanti will not attend unless Jasmine also does. Which combinations of these three friends can you invite so as not to make someone unhappy?
Are these system specifications consistent? “If the file
system is not locked, then new messages will be queued.
If the file system is not locked, then the system is func-
tioning normally, and conversely. If new messages are not
queued, then they will be sent to the message buffer. If
the file system is not locked, then new messages will be
sent to the message buffer. New messages will not be sent
to the message buffer.”
#339914 on Dec 2023