| № |
Condition |
free/or 0.5$ |
| m96166 | Provide the details for the proof of part (b) of Theorem 7.7.
Theorem 7.7
If A is a set, then
(b) Any partition of A gives rise to an equivalence relation R on A. |
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| m96167 | Provide the details for the proof of Theorem 3.6(b).
Theorem 3.6(b)
(b) |
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| m96168 | Provide the justifications (selected from the laws of set theory) for the steps that are needed to simplify the set
(A ∩ B) ∪ [B ∩ ((C ∩ D) ∪ (C ∩ ))],
where A, B, C, D ⊂ U. |
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| m96169 | Provide the missing reasons for the steps verifying the following argument:
Steps Reasons
(1) ∀x [p(x) ∨ q(x)] Premise
(2) ∃x ¬p(x) Premise
(3) ¬ p(a) Step (2) and the definition of the truth for ∃x ¬ p(x).
[Here a is an element (replacement) from the universe
for which ¬p(x) is true.] The reason for this step is also
referred to as the Rule of Existential Specification.
(4) p(a) ∨ q(a)
(5) q(a)
(6) ∀x [¬q(x) ∨ r(x)]
(7) ¬q(a) ∨ r(a)
(8) q(a) → r(a)
(9) r(a)
(10) ∀x [s(x) → ¬ r (x)]
(11) s(a) → ¬ r(a)
(12) r(a) → ¬s(a)
(13) ¬s(a)
(14) ∴ ∃x ¬s(x) Step (13) and the definition of the truth for ∃x ¬s(x).
The reason for this step is also referred to as the Rule
of Existential Generalization. |
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| m96170 | Provide the proofs for the remaining parts of Theorems 6.1 and 6.2. |
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| m96171 | Provide the proofs for Theorem 7.2(a), and (b).
Theorem 7.2
Given a set A with |A| = n and a relation R on A, let M denote the relation matrix for R. Then
(a) R is reflexive if and only if In ≤ M.
(b) R is symmetric if and only if M = Mtr. |
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| m96172 | Provide the reasons for the steps verifying the following argument. (Here a denotes a specific but arbitrarily chosen element from the given universe.)
Steps Reasons
(1) ∀x [p(x) → (q(x) ∧ r(x))]
(2) ∀x [p(x) ∧ s(x)]
(3) p(a) → (q(a) ∧ r(a))
(4) p(a) ∧ s(a)
(5) p(a)
(6) q(a) ∧ r(a)
(7) r(a)
(8) s(a)
(9) r(a) ∧ s(a)
(10) ∴ ∀x [r(x) ∧ s(x)] |
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| m96173 | Psychologist: places a rat each day in a cage with two doors, A and B. The rat can go through door A, where it receives an electric shock, or through door B, where it receives some food. A record is made of the door through which the rat passes. At the start of the experiment, on a Monday, the rat is equally likely to go through door A as through door B. After going through door A and receiving a shock, the probability of going through the same door on the next day is 0.3. After going through door B and receiving food, the probability of going through the same door on the next day is 0.6.
(a) Write the transition matrix for the Markov process.
(b) What is the probability of the rat going through door A on Thursday (the third day after starting the experiment)?
(c) What is the steady-state vector? |
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| m96236 | Recall that in a standard deck of 52 cards there are 12 picture cards - four each of jacks, queens, and kings. Kevin draws one card from the deck. Find the probability his card is a king if we know that the card drawn is an ace or a picture card. |
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| m96262 | Refer to the discussion following Example 3 to develop the double angle identities for sine and cosine by using the matrix transformation f(f(u)) = A(Au), where |
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| m96272 | Related to the merge sort is a somewhat more efficient procedure called the quick sort. Here we start with a list L: a1, a2, . . . , an, and use a1 as a pivot to develop two sublists L1 and L2 as follows. For i > 1, if al < a1, place ax at the end of the first list being developed (this is L1 at the end of the process); otherwise, place ax at the end of the second list L2.
After all al, i > 1, have been processed, place a1 at the end of the first list. Now apply quick sort recursively to each of the lists L1 and L2 to obtain sublists L11, L12, L21, and L22. Continue the process until each of the resulting sublists contains one element. The sublists are then ordered, and their concatenation gives the ordering sought for the original list L.
Apply quick sort to each list in Exercise 2. |
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| m96273 | Renu wants to sell her laptop for $4000. Narmada offers to buy it for $3000. Renu then splits the difference and asks for $3500. Narmada likewise splits the difference and makes a new offer of $3250.
(a) If the women continue this process (of asking prices and counteroffers), what will Narmada be willing to pay on her 5th offer? kth offer? kth offer, k ≥ 1?
(b) If the women continue this process (providing many, many new asking prices and counteroffers), what price will they approach?
(c) Suppose that Narmada was willing to buy the laptop for $3200. What should she have offered to pay Renu the first time? |
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| m96274 | Repeat Exercise 1 with "reflexive" replaced by
(i) Symmetric;
(ii) Antisymmetric;
(iii) Transitive.
Exercise 1
Let A be a set and I an index set where, for each i ∈ I, Ri is a relation on A. Prove or disprove each of the following.
(a)
is reflexive on A if and only if each Ri is reflexive on A.
(b)
is reflexive on A if and only if each Ri is reflexive on A. |
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| m96275 | Repeat Exercise 3 with Z5 taking the place of Z3. |
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| m96276 | Repeat Exercise 5 for each of the following linear systems:
(a) x + y + 2z + 3w = 13
x - 2y + z + w = 8
3x + y + z - w = 1
(b) x + y + z = 1
x + y - 2z = 3
2x + y + z = 2
(c) 2x + y + z - 2w = 1
3x - 2y + z - 6w = -2
x + y - z - w = - 1
6x + z - 9w = - 2
5x - y + 2z - 8w = 3 |
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| m96297 | Rework Example 15.20 using a Karnaugh map on six variables.
Example 15.20
For the graph shown in Fig. 15.9, let the vertices represent cities and the edges highways. We wish to build hospitals in some of these cities so that each city either has a hospital or is adjacent to a city that does. In how many ways can this be accomplished by building a minimal number of hospitals in each case? |
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| m96298 | Rewrite each of the following statements as an implication in the if-then form.
(a) Practicing her serve daily is a sufficient condition for Darci to have a good chance of winning the tennis tournament.
(b) Mary will be allowed on Larry s motorcycle only if she wears her helmet. |
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| m96299 | Rewrite each of the following statements in the if-then form. Then write the converse, inverse, and contrapositive of your implication. For each result in parts (a) and (c) give the truth value for the implication and the truth values for its converse, inverse, and contrapositive. [In part (a) "divisibility" requires a remainder of 0.]
(a) [The universe comprises all positive integers.] Divisibility by 21 is a sufficient condition for divisibility by 7.
(b) [The universe comprises all snakes presently slithering about the jungles of Asia.]
Being a cobra is a sufficient condition for a snake to be dangerous.
(c) [The universe consists of all complex numbers.]
For every complex number z, z being real is necessary for z2 to be real. |
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| m96303 | Russell draws one card from a standard deck. If A, B, C denote the events
A: The card is a spade.
B: The card is red.
C: The card is a picture card (that is, a jack, queen, or king).
Find Pr(A ∪ B ∪ C). |
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| m96326 | Sergeant Bueti must distribute 40 bullets (20 for rifles and 20 for handguns) among four police officers so that each officer gets at least two, but no more than seven, bullets of each type. In how many ways can he do this? |
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