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Paper Code: MATH502
Algebra and Discrete Mathematics
Assignment 2
Due 9:00am, Thursday, 14 May 2020
Name ........................................ Student Number ..........................
Stream Number .......
Question Marks Possible Marks Given
Instructions:
Please include this sheet as the first scanned page of your assignment. Submit completed
solutions via Blackboard, under the Assessment link.
Answer all questions and show your working. No working = no marks.
This is an individual assignment. The point of the assignment is for you to go through the process of
discovery for yourself. Copying someone else’s work will not achieve this. Plagiarism has occurred
where a person effectively and without acknowledgement presents as their own work the work of
others. That may include published material, such as books, newspapers, lecture notes or handouts,
material from the internet or other students’ written work.
The School of Computer and Mathematical Sciences regards any act of cheating including plagiarism,
unauthorised collaboration and theft of another student’s work most seriously. Any such act
will result in a mark of zero being given for this part of the assessment and may lead to disciplinary
action.
I understand the definition of plagiarism and I assert that this submission is my own work.
Signature: ........................................
Question 1 (12 marks). Let the Universal set, U = {x ∈ N | x < 10}, A = {x | 2 < x < 8}, and
B = {x | x > 4}. (Note that 0 ∈ N.) Show the following sets by enumeration:
a) A ∪ B
b) P(A ∩ B)
c) {X | X ⊆ (A ∩ B), 6 ∈ X}
d) {X | X ∩ (A ∪ B) = ∅}
Question 2 (16 marks). Let U, A, B, C be as indicated in the diagram below:
U
A B
C
Draw a Venn diagram and shade the area corresponding to each of the following. Show all work
including any intermediate Venn diagrams as required.
a) A ∩ (B ∪ C)
b) (A ∩ B) ∪ C
c) A ∩ (B ∪ C)
d) A ∪ (B ∩ C)
Question 3 (6 marks). Use the set identities on the last page to prove any results indicated by
the Venn Diagrams in Question 2.
Question 4 (12 marks). Let the universal set U = N ∪ P(N) and consider A = {0, 1, {2, 3, 4}},
B = ∅, and C = {1, 2}. What are the cardinalities of the following sets? Show all work.
a) A ∪ C
b) C − B
c) A × C
d) C × B
e) P(A ∩ B)
f) P(P(A))
2
Question 5 (16 marks). Consider the family of sets, Ak = {a ∈ N | a = k
r, r ∈ N} where the
indexing set S = {k ∈ N | k > 0}. Prove the following:
a) If m = ns, where s ∈ N, then Am ⊆ An.
b) The function fm,n : Am × An → Amn defined by fm,n(a, b) = ab is surjective.
Question 6 (20 marks). For each of the following relations, determine if f is
• reflexive,
• symmetric,
• antisymmetric, or
• transitive.
Conclude by stating if the relation is an equivalence, a partial order, or neither.
For each point, state your reasoning in proper sentences.
a) f = {(a, b) ∈ Z
2
| a ≡ b mod n, n ∈ Z}
b) f = {(X, Y ) ∈ P(A)
2
| X ⊂ Y }, where A is a non-empty set
Question 7 (18 marks). For each of the following relations, determine if f is
• a function,
• surjective, or
• injective.
Conclude by stating if the relation represents a bijective function.
For each point, state your reasoning in proper sentences.
a) f = {(a, b) ∈ N × N2| a ∈ N, b ∈ N2, b = (a, 2a)}
b) f = {(x, y) ∈ R2| y =√x}
Set Identities
A ∩ A = A Idempotent A ∪ A = A
A ∩ B = B ∩ A Commutative A ∪ B = B ∪ A
(A ∩ B) ∩ C = A ∩ (B ∩ C) Associative (A ∪ B) ∪ C = A ∪ (B ∪ C)
A ∩ (A ∪ B) = A Absorption A ∪ (A ∩ B) = A
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Distributive A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A = A Involution
A ∪ B = A ∩ B De Morgan’s A ∩ B = A ∪ B
A − B = A ∩ B Difference
A ∪ ∅ = A Identity A ∩ U = A
A ∪ U = U Domination A ∩ ∅ = ∅
A ∪ A = U Complement A ∩ A = ∅
3

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