2022-23-2
Discrete Mathematics
Assignment
Prepared by Dr. Patrick Chan
Total Marks: 76
1)
(10 marks)
Show if the following statements are logically equivalent. If no, provide a counterexample. DO
NOT use truth table.
a) (¬(p ↔ q))  (p  r) and T
b) (pq)  (pr) and p  (q  r)
2)
(12 marks)
Given the following variable definition:
P(x) : x works hard
Q(x): x is clever
S(x): x passes the discrete mathematics examination
Hypothesis:
1 Mary works hard or Peter is clever.
2
3
4
5
P(Mary)  Q(Peter)
Peter passes the discrete mathematics examination if and only if
Paul works hard.
Mary cannot pass the discrete mathematics examination.
S(Peter)  P(Paul)
If a person is clever, he/she can pass the discrete mathematics
examination.
If a person can pass the discrete mathematics, he/she works
hard.
x(Q(x) S(x))
S(Mary)
x(S(x)P(x))
Conclusions:
i) Paul passes the discrete mathematics examination
ii) Mary is not clever
a) Express the conclusions in terms of the variables above.
(4 marks)
b) Based on the hypothesis above, determine if each conclusion can be drawn. If yes, write down
the proof in detail. No truth table should be used.
(8 marks)
3)
(8 marks)
Prove or disprove that if n is a positive integer, then n is odd if and only if 5n + 6 is odd.
Discrete Mathematics – Assignment
1
4)
(10 marks)
Given f(x) = x2 + 5, g(x) = 2x + 1 and h(x) = | x |, are functions from R to R.
a) Range of h(x)
(2 marks)
b) Find g o f
(2 marks)
c) Find f + g
(2 marks)
d) Discuss the restricted domain and codomain in which g(x) is one-to-one correspondence.
(4 marks)
5)
(12 marks)
Let R1 and R2 be relations on a set A represented by the matrices
M R1
0 1 1
0 1 0 
 1 0 0 and M R2  0 0 1
0 1 1
0 1 1
Find the matrices that represent:
a) R1-1
b) R2 o R1
6)
c) transitive closure of R2
(12 marks)
Determine whether the following relation R whether it is i) symmetric, ii) antisymmetric, iii)
asymmetric, iv) transitive, v) irreflexive, vi) reflexive, vii) function, viii) invertible function, ix)
equivalence relation, x) partial order relation.
a) {(1,2), (6,1), (5,4), (2,3), (3,2), (4,5)} on set {1,2,3,4,5,6}
b) “Person A and Person B have the same mother” on the set of all persons
c) x2 ≥ y on the set of all real numbers
7)
(12 marks)
Given a relation R “divisibility” on the set {1, 2, 3, 4, 6, 9, 11, 12, 18}.
a) Draw the Hasse diagram
b) Find the minimal elements of {2, 3, 6, 9, 11, 12}
c) Find the least elements of {1, 2, 3, 6, 12}
d) Find the least upper bound of {1, 4}
e) Find the lower bounds of {4, 9}
Discrete Mathematics – Assignment
(4 marks)
(2 marks)
(2 marks)
(2 marks)
(2 marks)
2