ma2144a2
Sets, Relations and Functions
1. For each of the following, choose an appropriate universe of discourse and a predicate to define the set
and its complement.
(a) The set of all odd integers.
(b) The set of human fathers.
(c) the set of tautolo gi
MA2184/MA2504 Discrete Mathematics for Computing (08-09 sem B)
Take Home Quiz 1 (Due date: 6-3-2009)
1. Let F (x) be the predicate x is a freshman, M (y ) be the predicate y is a math course, B (x) be the predicate
x is a full-time student and T (x, y ) b
MA2184/MA2504 Discrete Mathematics for Computing (08-09 sem B)
Take Home Quiz 2 (Due date: 17-4-2009)
Suggested Solutions:
1. a) Since each element in A can map to any elements in B , so each of them has 10 ways to form a function,
thus, there are totally
MA2184/MA2504 Discrete Mathematics for Computing (08-09 sem B)
Take Home Quiz 2 (Due date: 17-4-2009)
1. Suppose |A| = 5 and |B | = 10.
a) Find the number of functions f : A B . (Give Reason)
(10 marks)
b) Find the number of injective functions f : A B .
A binary relation R on A is an equivalence relation if R is reflexive, symmetric and
transitive. An equivalence relation provides a means of treating the members of a set
according to their properties rather than as individuals.
Example
Let A = cfw_1,2,3,
Euler Paths
Question: Is it possible to start at some location in the town and travel across all the
bridges without crossing any bridge twice, and return to the starting point?
This is equivalent to the following.
Question: Is there a simple circuit in t
We will now discuss generating functions, which have been widely used. The idea behind the method is very
simple. Given a sequence, we aim to find a function, called the generating function of the sequence, so that
all useful properties of the sequence ca
ma2184a3, ma2504a3
Combinatorics
1. Using pigeon hole principal show that in any gathering of six people there are either three people who all
know each other or three people none of whom knows either of the other two.
2
(a) If A and B are finite sets wit
ma2184a5, ma2504a5
Graphs and Trees
1. Does there exist a simple graph with five vertices of the following degrees? If so, draw such a graph.
(a) 3, 3, 3, 3, 2
(b) 1, 2, 3, 4, 5
(c) 3, 4, 3, 4, 3
(d) 0, 1, 2, 2, 3
2.
How many vertices will the following g
ma2184a1 Logic
1. If p stands for it rains in Paris and q stands for we bring umbrellas, then express each of the
following in symbolic form.
(a) If it rains in Paris, then we bring our umbrellas.
(b) It does not rain in Paris and we bring umbrellas.
(c)
ma2184a2
Sets, Relations and Functions
1. For each of the following, choose an appropriate universe of discourse and a predicate to define the set
and its complement.
(a) The set of all odd integers.
(b) The set of human fathers.
(c) the set of tautolo gi
ma2184a3, ma2504a3
Combinatorics
1. Using pigeon hole principal show that in any gathering of six people there are either three people who all
know each other or three people none of whom knows either of the other two.
Solution:
Lets index 6 people, say a
ma2184a5, ma2504a5
Graphs
1. Does there exist a simple graph with five vertices of the following degrees? If so, draw such a graph.
(a) 3, 3, 3, 3, 2
(b) 1, 2, 3, 4, 5
(c) 3, 4, 3, 4, 3
(d) 0, 1, 2, 2, 3
Solution:
(a)
Yes,
(b)
No, sum of degrees is odd.
(
ma2184a1, ma2504a1
Logic
1. If p stands for it rains in Paris and q stands for we bring umbrellas, then express each of the
following in symbolic form
(a) if it rains in Paris, then we bring our umbrellas,
(b) it does not rain in Paris and we bring umbrel
MA2184/MA2504 Discrete Mathematics for Computing (08-09 sem B)
Take Home Quiz 1 (Due date: 6-3-2009)
Suggested Solutions:
1. a) x(B (x) F (x)
b) xy [(B (x) F (x) (M (y ) T (x, y )]
c) x[B (x) y (M (y ) T (x, y )]
2.
1 x(P (x) Q(x) p
2
x(P (x) Q(x)
3
P (c)