Islamic University of Gaza
Faculty of Engineering
Department of Computer Engineering
Fall 2011
ECOM 2311: Discrete Mathematics
Eng. Ahmed Abumarasa
Discrete Mathematics
Sec 1.1
The Foundations: Logic and Proof, Sets, and Functions
Logic
Chapter 1:
The Foundations: Logic and Proof, Sets, and Functions
 1.1: Logic


Propositions must have clearly defined truth values (True or False), so a proposition must
be a declarative sentence with no free variables.
Not: the negative of the propositions.
E.g. the negative of propositions p is (ꜚp called not p)

Conjunction: the conjunction between two proposition is the AND connector.
(pᶺq) is true only when both p and q are true.

Disjunction: the disjunction between two proposition is the OR connector.
(pᵛq) is false only when both p and q are false.

Exclusive OR:

Conditional Statements: p
o
o
o
q
CONVERSE: q p
CONTRAPOSITIVE: ꜚq
INVERSE: ꜚp ꜚq
ꜚp
 Biconditional:
(p ↔ q)p if and only if q.
 consistent System:
System is consistent if all its propositions are true.

Logic and Bit Operations:
Exercises:
1) 1.1.2: Which of these are propositions? What are the truth values of those that are
propositions?
Do not pass go.
What time is it?
There are no black flies in Maine
4+ X = 5
x + 1 = 5 if x = 1.
x + y = y + z if x = z.
Not a proposition
Not a proposition
a proposition
Not a proposition
Not a proposition
a proposition
2) 1.1.4: Let p and q be the propositions
p: I bought a lottery ticket this week.
q: I won the million dollar jackpot on Friday.
Express each of these propositions as an English sentence.
a
b
c
d
e
f
g
h
I did not buy a lottery ticket this week.
Either I bought a lottery ticket this week or I won the million dollar jackpot on Friday.
[in the inclusive sense]
If I bought a lottery ticket this week, then I won the million dollar jackpot on Friday.
I bought a lottery ticket this week and I won the million dollar jackpot on Friday.
I bought a lottery ticket this week if and only if I won the million dollar jackpot on Friday.
If I did not buy a lottery ticket this week, then I did not win the million dollar jackpot on
Friday.
I did not buy a lottery ticket this week, and I did not win the million dollar jackpot on Friday.
Either I did not buy a lottery ticket this week, or else I did buy one and won the million
dollar jackpot on Friday.
3) 1.1.9: Let p and q be the propositions
p : You drive over 65 miles per hour.
q : You get a speeding ticket.
Write these propositions using p and q and logical connectives.
a) You do not drive over 65 miles per hour.
b) You drive over 65 miles per hour, but you do not get a speeding ticket.
c) You will get a speeding ticket if you drive over 65 miles per hour.
d) If you do not drive over 65 miles per hour, then you will not get a speeding ticket.
e) Driving over 65 miles per hour is sufficient for getting a speeding ticket.
f) You get a speeding ticket, but you do not drive over 65 miles per hour.
g) Whenever you get a speeding ticket, you are driving over 65 miles per hour.
a
Not p
b
pᶺ (not p)
c
p
d
not p
q
not q
e
p
f
qᶺ (not p)
q
4) 1.1.9: Construct a truth table for each of these compound propositions.

A
p
T
F

P
T
T
F
F
output
F
F
C
P
T
T
F
F

not p
F
T
q
T
F
T
F
not q
F
T
F
T
P v not q
T
T
F
T
Output
T
F
T
F
E
q
T
F
T
F
not p
F
F
T
T
not q
F
T
F
T
p
T
F
T
T
q
not q
T
F
T
T
not p
Output
T
T
T
T
g
q
p