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250 members of a certain society have voted to elect a new chairman. Each member may vote for either one or two candidates. The candidate elected is the one who polls most votes. Three candidates x, y z stood for election and when the votes were counted, it was found that: - 59 voted for y only, 37 voted for z only - 12 voted for x and y, 14 voted for x and z - 147 voted for either x or y or both x and y but not for z - 102 voted for y or z or both but not for x Required i. Present the information in a Venn diagram. (6 Marks) ii. How many voters did not vote? (4 Marks) iii. How many voters voted for x only? (2 Marks) iv. Who won the elections?
whether it is the linearly ordered set or not.
Let A = $1, 2, 3, 4, 6, 9} and let R be the relation on A defined by "x divides y "
written x/y.
a) Write R as a set of ordered pairs
b) Drawits directed graph.
c) Find indegree & outdegree of each vertex
d) Write the relation matrix of it.
) Find the inverse relation of R.
250 members of a certain society have voted to elect a new chairman. Each member may
vote for either one or two candidates. The candidate elected is the one who polls most votes.
Three candidates x, y z stood for election and when the votes were counted, it was found that:
- 59 voted for y only, 37 voted for z only
- 12 voted for x and y, 14 voted for x and z
- 147 voted for either x or y or both x and y but not for z
- 102 voted for y or z or both but not for x
Required
i. Present the information in a Venn diagram. (6 Marks)
ii. How many voters did not vote? (4 Marks)
iii. How many voters voted for x only? (2 Marks)
iv. Who won the elections? (2 Marks)
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 ?
