What is discrete mathematics?
Discrete mathematics is devoted to the study of discrete or
distinct unconnected objects.
Classical mathematics deals with functions on real numbers.
Real numbers form a continuous line.
Some calculus techniques apply only to

An application of propositional logic
Consider the following kids Sudoku puzzle:
The rules are:
1. Each row must contain all the numbers 1-4.
2. Each column must contain all the numbers 1-4.
3. Each 2 2 block must contain all the numbers 1-4.
4. No cell c

Discrete Mathematics
Discrete mathematics is devoted to the study
of discrete or distinct unconnected objects.
Classical mathematics deals with functions
on real numbers. Real numbers form a
continuous line. Some calculus techniques
apply only to contin

Propositional Equivalences
A tautology is a compound proposition that is
True for all truth values of the atomic propositions in it. Example:
p p
A contradiction is a compound proposition
that is False for all truth values of the atomic
propositions in it

Predicates and Quantiers
Statements involving variables, like
x2 x + 2
The American city is polluted.
are not propositions, since their truth values
depend on the values of the variable involved.
For what value of x?
Which city?
1
Nevertheless, we sometim

Where is the treasure?
f the house is next to the lake, then the treasure is not in the kitchen
f the tree in the front yard is an elm, then the treasure is in the kitchen
he house is next to the lake
he tree in the front yard is an elm or the treasure is

able x ranges over all English texts.
means x is a clear explanation
means x is satisfactory
means x is an excuse
ress the following in predicate calculus:
ll clear explanations are satisfactory
ome excuses are unsatisfactory
ome excuses are not clear exp

Concordia University
Department of Computer Science & Software Engineering
COMP 232/4
Mathematics for Computer Science
Fall 2014
Sample of midterm questions for COMP 232/Q
1. What are the converse and contrapositive of the statement If it is sunny, then I

COMP 232 Mathematics for Computer Science
Fall 2014
Midterm Exam II
Total Points:
Name:
/ 30
ID:
Instructions. This is a closed book exam. The only allowed tool is an ENCS approved calculator. Provide all answers in this booklet. Use pen, not pencil. Do n

COMP 232 Mathematics for Computer Science
Fall 2014
Midterm Exam
Total Points:
Name:
/ 20
ID:
Instructions. This is a closed book exam. The only allowed tool is an ENCS approved calculator. Provide all answers in this booklet. Use pen, not pencil. Do not

Injective functions
A function f A B is one-to-one, or injective if for any two distinct elements x, y A we
have f (x) f (y). IOW f is one-to one =def
x A, y A (x y f (x) f (y)
Example: f1 N N, where f1(n) = 2n + 1.
If x y then 2x + 1 2y + 1. Thus, f1 is

Concordia University
Department of Computer Science
and Software Engineering
COMP 232
Mathematics for Computer Science
Fall 2014
Assignment 3
Due: November 4 at midnight
1. Are the following equalities valid for all integers n and m ? If yes then give a p

Concordia University Department of Computer Science
COMP 232
Mathematics for Computer Science
Fall 2014
Solutions to Assignment 3
1. Are the following equalities valid for all integers n and m ? If yes then give a proof. If no
then give a counterexample.