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Discrete Mathematics
Revision Questions of Topic 5 - 7
Topic 5: Propositional Logic
1.
Show that the following compound propositions are logically equivalent by using the
truth table.
~ (~ p  q)  ( p  q)  p
2.
Given the three propositions as follows.
p: Today is holiday.
q: I go to college.
r: I play tennis.
Represent each statement below using p, q, r and logical connectives.
(a)
If today is not holiday, then I go to college or I play tennis.
(b)
I play tennis if and only if today is holiday.
(c)
Today is not holiday but I don’t go to college.
3.
Write the negation of the statement below.
Not all the students take Calculus.
4.
Write the disjunction of the given pairs of statements p and q below, then determine
whether it is true or false.
P: One is an even integer.
q: Nine is a positive integer.
5.
Construct a truth table for the compound proposition given below and determine
whether it is tautology, contradiction or indeterminant.
(q ~ r ) → ( p  r )
6.
Given the three propositions as follows.
p: I get a bonus.
q: I will sell my motorcycle.
r: I will buy a car.
Write a sentence for each of the following:
(a)
p→r
(b)
~ p → (~ q ~ r )
(c)
(r  q)  p
7.
Determine whether the following statements are true or false.
(a)
1 is not a prime number and 1 is not the smallest positive integer.
(b)
3 is even number or 3 + 2 < 8.
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Topic 6: Boolean Algebra
1.
Given a logic circuit as follows:
x
y
F
z
(a)
(b)
2.
Find the Boolean expression that represents the output F of the circuit above.
Construct a truth table for the expression obtained in part (a).
Obtain the simplified Boolean function from the Karnaugh map below. Show all your
workings.
CD
AB
00
01
11
1
1
1
01
1
1
1
11
1
1
1
10
1
1
00
3.
10
Given a truth table of a Boolean Function as follows.
x
0
0
0
0
1
1
1
1
(a)
(b)
4.
y
0
0
1
1
0
0
1
1
z
0
1
0
1
0
1
0
1
Output
0
1
1
0
1
1
1
0
State the minterm and maxterm expressions of the outputs given in the table.
Simplify the minterm expression in part (a) using Karnaugh Map. Give all the
two possible answers of the simplified minterm expression.
Obtain the simplified Boolean function from the Karnaugh map below. Show all your
workings.
CD
AB
00
01
11
10
00
1
1
01
1
1
11
10
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1
1
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5.
Construct a circuit for the output below.
6.
Given a truth table of a Boolean Function as follows.
(xy'+ z )(x + y )'
x
0
0
0
0
1
1
1
1
(a)
(b)
y
0
0
1
1
0
0
1
1
z
0
1
0
1
0
1
0
1
Output
1
1
1
0
0
0
1
1
State the minterm and maxterm expression of the outputs given in the table.
Simplify the minterm expressions in part (a) by using Karnaugh Map. Give all
the two possible answers of simplified minterm expression.
Topic 7: Graphs & Trees
1.
Given a simple graph with a degree sequence of (2, 2, 3, 3, 4).
(a)
How many edges are there in the graph?
(b)
Draw a simple graph with the degree given above.
2.
Given a rooted tree as follows:
a
b
c
d
e
i
h
(a)
(b)
(c)
3.
g
f
j
k
What is the height of the tree above?
List all the internal vertices of the tree.
List the order of the vertices for the tree if it follows preorder traversal.
Find the in-degree and out-degree for all the vertices of the directed graph below.
B
C
A
D
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E
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4.
Given an undirected graph as follows:
b
a
c
e
d
(a)
(b)
Represent the graph using adjacency list.
Discuss whether the graph contains a Euler circuit or Euler path. Construct such
a path or circuit when one exists.
Discuss whether the graph contains a Hamilton circuit. If it does, find such a
circuit.
(c)
5.
6.
(a)
Draw an undirected graph based on the set of vertices, V and set of edges, E
given below.
V = {a, b, c, d}
E = {{a, b}, {a, c}, {b, d}, {b, c}, {c, d}}
(b)
(c)
Find the degree of all vertices.
Give reason why Euler circuit does not exist.
Given a directed graph as follows:
a
b
c
d
7.
(a)
Represent the directed graph above by using adjacency list.
(b)
Find the in-degree of all the vertices.
(c)
Find a simple circuit with length of 4.
Given a rooted tree as follows.
a
b
c
d
h
(a)
(b)
(c)
(d)
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e
i
j
g
f
k
l
What are the vertices at level 2?
What is the parent of f and g?
How many leaves are there of the rooted tree above?
List all the ancestors of j.
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(e)
(f)
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What is the sibling of e?
List the order of the vertices of the tree if it follows
(i)
preorder traversal
(ii)
postorder traversal
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