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COS1501 EXAM PACK 2022
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COS1501 - Theoretical Computer Science I Summary Study Notes
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COS1501EXAM PACK
, 2
COS1501
Example examination paper with discussions
SECTION 1
SETS AND RELATIONS
(Questions 1 to 12) (12 marks)
Questions 1 to 8 relate to the following sets:
Suppose U = {1, {2, 3}, 3, d, {d, e}, e} is a universal set with the following subsets:
A = {{2, 3}, 3, {d, e}}, B = {1, {2, 3}, d, e} and C = {1, 3, d, e}.
Before attempting the questions, let us write down the sets U, A, B and C, by adding spaces
between elements, so that common elements are vertically grouped:
U = {1, {2, 3}, 3, d, {d, e}, e}
A = { {2, 3}, 3, {d, e}}
B = {1, {2, 3}, d, e}
C = {1, 3, d, e}
We can clearly see, for example, that element {d, e} in U appears in subset A only. Or that
elements 1, d and e in U, also appear in subsets B and C. Or that the intersection of A and C
contains element 3 only. If you find it difficult to see which elements are in which set, it may
help you to write it in this way on rough in the exam.
Question 1
Which one of the following sets represents A B?
1. {1, 2, 3, d, e}
2. {1, {2, 3}, 3, d, e}
3. {1, {2, 3}, 3, {d, e}}
4. {1, {2, 3}, 3, {d, e}, d, e}
Discussion:
A = { {2, 3}, 3, {d, e}}
B = {1, {2, 3}, d, e}
A B represents the union of the sets A and B. This means, it contains elements that are in A
or in B or in both A and B. (Study guide p. 41).
Thus A B = {1, {2, 3}, 3, d, {d, e}, e}. This corresponds to alternative 4. Remember that the
order of the elements in the set does not matter, as long as all elements are in the set. We
also do not repeat the same element in the set – although the element {2, 3} is in both A and
B, it only appears once in the set A B.
[TURN OVER]
, 3
COS1501
Example examination paper with discussions
Question 2
Which one of the following sets represents B C?
1. {1, 3, d, e}
2. {1, d, e}
3. {d, e}
4. {3, {2, 3}}
Discussion:
B = {1, {2, 3}, d, e}
C = {1, 3, d, e}
The intersection of B and C contains all the elements that are in both subsets B and C, ie
elements that are common in B and C (Study guide p. 42).
Thus B C = {1, d, e}, corresponding to alternative 2.
Question 3
Which one of the following sets represents C – A?
1. {1, {2, 3}, d, e, {d, e}}
2. {3, d, e}
3 {}
4. {1, d, e}
Discussion:
A = { {2, 3}, 3, {d, e}}
C = {1, 3, d, e}
C – A (Set difference / C without A) is the set of all elements that are in C, but not in A (Study
guide p. 42). This means that if an element appears in both A and C, it will be removed from
C to get C – A. It is clear that element 3 is in both A and C and should be removed from C,
thus
C – A = {1, d, e}, corresponding to alternative 4.
[TURN OVER]
, 4
COS1501
Example examination paper with discussions
Question 4
Which one of the following sets represents U + B?
1. U
2. {3, {d, e}}
3. {1, {2, 3}, d, e}
4. (U – A) – C
Discussion:
U = {1, {2, 3}, 3, d, {d, e}, e}
B = {1, {2, 3}, d, e}
U + B (symmetric set difference) is the set of elements that are in U or in B, but not in both
(Study guide p. 43). This means we have to remove the elements that are in both U and B,
which are elements 1, {2, 3}, d and e. We then remain with elements 3 and {d, e}, thus
U + B = {3, {d, e}}, which corresponds to alternative 2.
Question 5
Which one of the following sets represents C B ?
1. {3}
2. {1, d, e}
3. {1, 3, d, e}
4. {1, 3, {d, e}}
Discussion:
U = {1, {2, 3}, 3, d, {d, e}, e}
B = {1, {2, 3}, d, e}
C = {1, 3, d, e}
We first determine B(the complement of B – study guide p. 42). The complement of B is the
set of all elements that is in U but not in B. From the above it is clear that if we remove the
elements 1, {2, 3}, d and e in B from U, we are left with elements 3 and {d, e}, therefore,
B = {3 , {d, e}}. Now we can determine which elements should be in C B.
C B = {1, 3, d, e} {3, {d, e}} = {3}, corresponding to alternative 1. (See definition of
intersection in Study guide p.42).
.
[TURN OVER]
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