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Logic and Set Theory Assignment





THE ATTACHED TEXT IS TAKEN DIRECTLY FROM THE WORD DOCUMENT CONTAINING THE QUESTIONS

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Question 1
(a) Consider the following argument.
If bears are brown then rabbits are not red. If giraffes are not green then
rabbits are red. Giraffes are green. Therefore bears are brown.
Define the following propositions:
b: bears are brown
g: giraffes are green
r: rabbits are red
(i) Rewrite the argument in symbols.
(ii) Construct a truth table for each premise and the conclusion.
(iii) State whether or not the argument is valid, and explain your answer.
(b) The propositions below can be arranged in three groups so that each member of a
group is logically equivalent to the other two. Find the three groups.
Give your answer by listing the three groups, for example "2, 5 and 9" (which is not
necessarily a correct answer)
[ Hint: It may help to realise that the first three of the expressions
p ? q, ~q ? ~p, ~p ? q, q ? p
are logically equivalent to each other but they are not logically equivalent
to the fourth (as you could verify using a truth table ). ]
[5 + 3 = 8 marks ]


Question 2
A group of 140 students is polled to see how many watched three TV shows, Awesome,
Bongo and Crikey. The results showed that 66 watched Awesome, 63 watched Bongo, 63
watched Crikey, 34 watched Awesome and Bongo, 31 watched Awesome and Crikey, 26
1 If the system is not ready then the light is not on.
2 Either the system is not ready or the light is not on.
3 If the system is ready then the light is on.
4 Either the system is not ready or the light is on.
5 If the light is on then the system is ready.
6 If the light is on then the system is not ready.
7 If the system is ready then the light is not on.
8 Either the light is not on or the system is ready.
9 If the light is not on then the system is not ready.
watched Bongo and Crikey, and 27 did not watch any of the three. Let A denote the set of
students who watched Awesome, and similarly define sets B and C.
(a) Calculate the number of students in each of the eight subsets shown in the Venn
diagram. Copy the Venn diagram and enter the number of students in each subset.
(b) Hence find how many students watched:
(i) Awesome and Crikey, but not Bongo;
(ii) Bongo only;
(iii) only two of the three shows;
(iv) at least two of the shows.
[ 8 + 2 = 10 marks ]


Question 3
Suppose that P is a set of people and M a set of movies. Define the predicate S(p, m) to
mean "person p has seen movie m". Consider the following list of sentences.
For each sentence, the negation is also in the list. Your task is to match each sentence with
its negation. Give your answer by listing the number of each sentence and its negation, for
example "4 and 10" (which is not necessarily a correct answer.)
[6 marks ]
?
A B
C
? ? ?
?
?
? ?
1 Everybody has seen all of the movies.
2 There is some person who has not seen any of the movies.
3 There is some movie that nobody has seen.
4 There is at least one movie that everybody has seen.
5 Nobody has seen any of the movies.
6 There is at least one movie that some person has not seen.
7 There is some person who has seen all the movies.
8 For each movie there is somebody who has not seen it.
9 Everybody has seen at least one movie.
10 For each person there is some movie that they have not seen.
11 For each movie there is somebody who has seen it.
12 There is at least one movie that some person has seen.


Question 4
Introduction:
There is a simple way of representing sets using so-called "bit strings". A bit string is simply
a string of 0's and 1's. Suppose that U is the universal set. Then the rule for the bit string for a
particular set S is:
If the kth element of U is in S, the kth bit of the bit string is 1.
If the kth element of U is not in S, the kth bit of the bit string is 0.
(Usually the order of the elements of a set is irrelevant. However when using bit strings the
order of the elements as specified in the universal set U must be adhered to.)
Example:
Suppose the universal set is U = {1, 2, 3, 4, …, 10}. Then the set A = {1, 3, 5, 9} is
represented by the bit string 1010100010 because:
the 1st element of U is in A, so the 1st element of the bit string is 1;
the 2nd element of U is not in A, so the 2nd element of the bit string is 0, and so on.
Similarly, the set {5, 6, 7} is represented by the bit string 0000111000.
Conversely, the bit string 0110010001 represents the set C = {2, 3, 6, 10} because:
the 1st element of the bit string is 0, so the 1st element of U, that is 1, is not in C;
the 2nd element of the bit string is 1, so the 2nd element of U, that is 2, is in C;
and so on.
Similarly, the bit string 0000000110 represents {8, 9}, and the bit string 0101010101
represents {2, 4, 6, 8, 10}.
The operations of union, intersection and complement may be carried out on the bit strings.
The simplest way is to use a table and to work "bitwise", that is carry out the operations on
the first bits, then the second bits, and so on.
Example:
Let the universal set be U = {4, 6, 9, 13, 18, 25}, and consider the sets A = { 6, 13, 18 } and
B = {4, 13, 18, 25}. Suppose that we want to find A ? B, A n B and A'. Using a table
and performing the operations "bitwise":
U 4 6 9 13 18 25
A 0 1 0 1 1 0
B 1 0 0 1 1 1
A ? B 1 1 0 1 1 1
A n B 0 0 0 1 1 0
A' 1 0 1 0 0 1
Then, from the table,
A ? B is represented by: 110111, so A ? B = {4, 6,13, 18, 25}
A n B is represented by: 000110, so A n B = {13, 18}
A' is represented by: 101001, so A' = {4, 9, 25}
Your task:

Let the universal set U be the following set of 16 countries:
U = {Angola, Benin, Chad, Djibouti, Eritrea, Ghana, Kenya, Libya, Mali,
Namibia, Rwanda, Sudan, Tanzania, Uganda, Zaire, Zambia}
(i) Find the bit string that represents the set L = { Benin, Chad, Libya, Tanzania }
(ii) Find the set M represented by the bit string 0010 1100 0011 0001
(iii) Let P, Q, R and S be sets represented by the following four bit strings respectively:
P 0 1 1 1 0 1 0 1 0 1 1 0 1 0 0 1
Q 0 1 0 1 0 0 1 1 1 0 1 0 1 0 1 1
R 1 1 0 1 1 1 0 0 0 1 0 0 1 1 1 0
S 1 1 0 1 0 1 1 0 1 0 0 1 1 0 0 1
By using bitwise operations on the four bit strings, find the bit string that represents
the set T = (P' n Q) ? (R n S'). (Set out your working in a table.)
(iv) List the countries in the set T defined in (iii).
[ 1+ 1 + 3 + 1 = 6 marks ]



Question 5
In the lecture notes there is an application of Karnaugh maps to a 7-segment display that is
used in some calculators and similar digital devices. The display is sometimes extended to
include all hexadecimal digits. A typical extended display is:
Notice the display for the hexadecimal digits for 10 through 15. Some are shown as upper
case and some lower case: A b c d E F. As in the lecture notes we label each of the
seven segments as shown:
When any one of the keys 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E or F is pressed, a 4-bit
binary signal "wxyz" is generated, as shown in the table
a
b
d
e c
f
g
6
Page 6 of 6
For each of the seven segments there is a corresponding Boolean function and a circuit that
has output 0 or 1 as the various keys are pressed. Denote these functions by a(w, x, y, z),
b(w, x, y, z), …, g(w, x, y, z).
Your task concerns the two segments d and g, and is as follows. For each of the two
functions d(w, x, y, z) and g(w, x, y, z):
(i) construct a truth table for the function (like the one shown above);
(ii) construct the corresponding Karnaugh map;
(iii) find a minimal sum of products for the function.
NOTE: Use the "simple" labelling on the Karnaugh maps and show your groups clearly.
[ 10 marks ]
Key Binary representation Segment Fundamental
w x y z … product
0 0 0 0 0
1 0 0 0 1
2 0 0 1 0
3 0 0 1 1
4 0 1 0 0
5 0 1 0 1
6 0 1 1 0
7 0 1 1 1
8 1 0 0 0
9 1 0 0 1
A 1 0 1 0
B 1 0 1 1
C 1 1 0 0
D 1 1 0 1
E 1 1 1 0
F 1 1 1 1


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Other Details about the Project/Assignment
Subjects: Mathematics -> Advanced Concepts
Topic: Logics, Sets, Karnuagh Maps, Boolean Logic
Level: College / University
Tags:

Logics, Sets, Karnuagh Maps, Boolean Logic


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Project Details
Subjects: Mathematics -> Advanced Concepts
Topic: Logics, Sets, Karnuagh Maps, Boolean Logic
Level: College / University
Tags:

Logics, Sets, Karnuagh Maps, Boolean Logic


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