Exam (elaborations) COS1501 Assignment 2 (COMPLETE ANSWERS) 2024 (653506) - 14 June 2024 • Course • Theoretical Computer Science I (COS1501) • Institution • University Of South Africa (Unisa) ...
Exam (elaborations) COS1501 Assignment 2 (COMPLETE ANSWERS) 2024 (653506) - 14 June 2024 • Course • Theoretical Computer Science I (COS1501) • Institution • University Of South Africa (Unisa) ...
Exam (elaborations) COS1501 Assignment 2 (COMPLETE ANSWERS) 2024 (653506) - 14 June 2024 • Course • Theoretical Computer Science I (COS1501) • Institution • University Of South Africa (Unisa) ...
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COS1501 Assignment
2 (COMPLETE
ANSWERS) 2024
(653506) - 14 June
2024; 100% TRUSTED
workings,
explanations and
solutions.
ADMIN
[COMPANY NAME]
,Question 1 Complete Marked out of 2.00 Let A, B and C be subsets of a
universal set U. Which one of the following four Venn diagrams
presents the set [(A ⋂ B) ’ – C] ⋂ [( A + B) – C ] ? (Hint: Draw the
Venn diagrams for [(A ⋂ B) ’ – C] ⋂ [( A + B) – C ] step by step). a. b.
c. d. Question 2 Complete Marked out of 2.00 Question 3 Complete
Marked out of 2.00 Let A, B and C be subsets of a universal set U = {1,
2, 3, 4}. The statement (A – B) U C’ = (C’ – B) + A is NOT an identity.
Which of the following sets A, B and C can be used in a counterexample
to prove that the given statement is not an identity? (Hint: substitute the
given sets for LHS and RHS separately) a. A = {1}, B = {2} and C =
{3} b. A = {1}, B = {1} and C = {2} c. A = {1, 2}, B = {1, 2} and C =
{3} d. A = {3}, B = {3, 4} and C = {4} We want to prove that for all A,
B, C ⊆ U, (A ⋂ B) U (C – B) = (A U C) ⋂ (A U B’) ⋂ (B U C) is an
identity. Consider the following incomplete proof: z ∈ ( A ⋂ B) U (C –
B) iff (z ∈ A and z ∈ B) or ( z ∈ C and z ∉ B) iff (z ∈ A or z ∈ C) and ( z
∈ A or z ∉ B) and (z ∈ B or z ∈ C) and ( z ∈ B or z ∉ B) iff Step 4 iff z
∈ (A U C) and z ∈ (A U B ’ ) and z ∈ (B U C) and z ∈ (B U B ’ ) iff
Step 6 iff z ∈ (A U C) ⋂ (A U B ’ ) ⋂ (B U C) ⋂ U iff z ∈ (A U C) ⋂
(A U B ’ ) ⋂ (B U C) [For any sets U and G, (G ⋂ U) = G.] Which one
of the following alternatives provides valid steps 4 and 6 to complete the
given proof? a. Step 4: iff (z ∈ A or z ∈ C) and (z ∈ A or z ∈ B’) and (z
∈ B or z ∈ C) and (z ∈ B or z ∈ B’ ) Step 6: iff z ∈ (A or C) and z ∈ (A
or B’) and z ∈ (B or C) and z ∈ U b. Step 4: iff (z ∈ A or z ∈ C) and (z ∈
A or z ∈ B’) and (z ∈ B or z ∈ C) and (z ∈ B or z ∈ B’) Step 6: iff z ∈ (A
U C) and z ∈ (A U B’) and z ∈ (B U C) and z ∈ U c. Step 4: iff (z ∈ A or
z ∈ C) and (z ∈ A or z ∉ B’) and (z ∈ B or z ∈ C) and (z ∈ B or z ∉ B’)
Step 6: iff z ∈ (A U C) or z ∈ (A U B’) or z ∈ (B U C) or z ∈ U d. Step
4: iff (z ∈ A and z ∈ C) or (z ∈ A and z ∈ B’) or (z ∈ B and z ∈ C) or (z
∈ B and z ∈ B’) Step 6: iff z ∈ (A U C) and z ∈ (A U B’) and z ∈ (B U
C) and z ∈ U Question 4 Complete Marked out of 2.00 Question 5
Complete Marked out of 2.00 Forty (40) students go to a party wearing
red, white and blue. Of these students, 17 wear red, 22 wear white, 25
, wear blue. (Students do not necessarily wear only one colour.)
Furthermore, 7 wear red and white, 12 wear blue and white, and 9 wear
red and blue . Which one of the following alternatives is true? (Hint:
Draw the Venn diagram, fi ll in the details given, and then fi rst calculate
the value of x, the unknown) a. 5 students wear red only. 8 students wear
white and blue, but not red. 3 students wear red and white, but not blue.
b. 2 students wear red only. 11 students wear white and blue, but not red.
6 students wear red and white, but not blue. c. 2 students wear red only.
8 students wear white and blue, but not red. 3 students wear red and
white, but not blue. d. 5 students wear red only. 11 students wear white
and blue, but not red. 6 students wear red and white, but not blue. Let T
be a relation from A = {0, 1, 2, 3} to B = {0, 1, 2, 3, 4} such that (a, b) ∈
T iff b – a is an odd number. (A, B ⊆ U = Z.) (Hint: Write down all the
elements of T. For example, if 4 ∈ B and 1 ∈ A then 42 – 12 = 16 – 1 =
15 which is an odd number,thus (1, 4) ∈ T.) Which one of the following
alternatives provides only elements belonging to T? a. (3, 1), (4, 1), (3,
2) b. (0, 1), (2, 4), (2, 3) c. (3, 0), (1, 2), (3, 4) d. (1, 0), (1, 2), (1, 3) 2 2
Question 1
To determine which Venn diagram represents the set
[(A∩B)′−C]∩[(A∪B)−C][(A \cap B)' - C] \cap [(A \cup B) -
C][(A∩B)′−C]∩[(A∪B)−C], let's break it down step by step.
1. (A∩B)′(A \cap B)'(A∩B)′ is the complement of A∩BA \cap
BA∩B.
2. (A∩B)′−C(A \cap B)' - C(A∩B)′−C is the set of elements in
(A∩B)′(A \cap B)'(A∩B)′ that are not in CCC.
3. (A∪B)−C(A \cup B) - C(A∪B)−C is the set of elements in A∪BA
\cup BA∪B that are not in CCC.
4. Finally, we find the intersection of the two sets from steps 2 and 3.
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