COS1501
Assignment 2
(COMPLETE
ANSWERS)
2024 (653506) -
14 June 2024
CONTACT:
, 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
Assignment 2
(COMPLETE
ANSWERS)
2024 (653506) -
14 June 2024
CONTACT:
, 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
