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Let D = {-5, -3, -1, 1, 3, 5}. Write the following statements using only negations, conjunctions and
disjunctions:
a) βπ₯π(π₯)
b) βπ₯π(π₯)
c) βπ₯((π₯ β 1) β π(π₯))
d) βπ₯((π₯ β₯ 0) β§ π(π₯))
e) βπ₯(οΏ’π(π₯)) β§ βπ₯((π₯ < 0) β π(π₯))
Write the negation of the following statement:
βπ₯βπ¦(π₯ + π¦ = 2 β§ 2π₯ β π¦ = 1
What rule of inference is used in each of these argu-
ments?
a) Alice is a mathematics major. Therefore, Alice is ei-
ther a mathematics major or a computer science major.
b) Jerry is a mathematics major and a computer science
major. Therefore, Jerry is a mathematics major.
c) If it is rainy, then the pool will be closed. It is rainy.
Therefore, the pool is closed.
d) If it snows today, the university will close. The uni-
versity is not closed today. Therefore, it did not snow
today.
e) If I go swimming, then I will stay in the sun too long.
If I stay in the sun too long, then I will sunburn. There-
fore, if I go swimming, then I will sunburn
Model the following situations as weighted graphs. Draw each graph, and complete the table below by filling in its direct neighbour. Leave the indirect neighbours as 0. It is well-known that in the Netherlands, there is a 2-lane highway from Amsterdam to Breda, another 2-lane highway from Amsterdam to Cappele aan den IJssel, a 3-lane highway from Breda to Dordrecht, a 1-lane road from Breda to Ede and another one from Dordrecht to Ede, and a 5-lane superhighway from Cappele aan den IJssel to Ede.
Prove by mathematical induction that n^2 +n < 2^n whenever n is an integer greater than 4.
Let D=(3,5,7,9) E=( 0, 4, 6, 9) F=0,3,6,7 . Universal set U = 0,1,2,3,4,5,6,7,8,9.
Draw venn diagram for.
(I) DnEnF
(II) (DUE) Uf
Consider all strings of length 12, consisting of all uppercase letters. Letters may be repeated. Please do not simplify your answers.
(a) How many such strings are there?
(b) How many such strings contain the word βSCOOBYβ?
(c) How many such strings contain neither the word βSCOOBYβ nor the word βDAPHNEβ?
Refer to a group of 191 students, of which 10 are taking math, business, and language; 36 are taking math and business; 20 are taking math and language; 18 are taking business and language; 65 are taking math; 76 are taking business and 63 are taking language.
26-30. Illustrate the Venn Diagram
Prove or disprove that if R and S are antisymmetric, then so is:
(a) (R βͺ S)
(b) (R β© S)
In Monopoly, your token is allowed to leave the βjailβ cell if you roll doubles: you roll two 6 -sided dice and each shows the same face. Zach hates being in jail, So he invents a couple of weighted dice that are not independent. In particular, if you roll either die on its own itβs a fair die: each outcome has probability 1/6. But if you roll one die and then the other, the red die will take the same outcome as the blue die exactly half the time: all other outcomes are equally likely.
(a) Suppose you roll a 3 on the blue die. What is the probability distribution of the red die given this outcome on the blue die?
(b) What is the probability you roll doubles?
(c) What is the probability that you roll a 7 as the sum of the two dice?
