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8/11/2019 Discrete Mathematics - Lecture 1 - Propositions
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Lecture 1: Propositions and Compound Propositions P a g e | 1
Discrete Structure / Mathematics
Lecture 1: PROPOSITIONS and COMPOUND
PROPOSITIONS
A proposition is any meaningful statement that is
either true or false, but not both. We will use
lowercase letters, such as p; q; r; ; to represent
propositions. We will also use the notation
p : 1 + 1 = 3
to define p to be the proposition 1+1 = 3: The truth
value of a proposition is true, denoted by T (or 1), if it
is a true statement and false, denoted by F (or 0), if it is
a false statement. Statements that are not
propositions include questions and commands.
Example 1.1
Which of the following are propositions? Give the truth
value of the propositions.a. 2 + 3 = 7
b. Julius Caesar was president of the United States.
c. What time is it?
d. Be quiet!
Solution.
a. A proposition with truth value (F).
b. A proposition with truth value (F).
c. Not a proposition since no truth value can be
assigned to this statement.
d. Not a proposition
Example 1.2
Which of the following are propositions? Give the truth
value of the propositions.
a. The difference of two primes.
b. 2 + 2 = 4:
c. Quezon City is the capital of the Philippines.
d. How are you?
Solution.
a. Not a proposition.
b. A proposition with truth value (T).
c. A proposition with truth value (F).
d.
Not a proposition.
New propositions called compound propositions or
propositional functionscan be obtained from old ones
by using symbolic connectives which we discuss next.
The propositions that form a propositional function are
called the propositional variables.
CONJUNCTION AND DISJUNCTION
Let p and q be propositions. The conjunctionof p and
q; denoted p ^ q; is the proposition: p and q: This
proposition is defied to be true only when both p and q
are true and it is false otherwise.
The disjunction of p and q; denoted p q; is the
proposition: p or q: The 'or' is used in an inclusive way.
This proposition is false only when both p and q are
false, otherwise it is true.
A truth table displays the relationships between thetruth values of propositions. Next, we display the truth
tables of p q and p q:
Example 1.3
Let
p : 5 < 9
q : 9 < 7
Construct the propositions p q and p q
Solution.
The conjunction of the propositions p and q is the
proposition:
p q : 5 < 9 and 9 < 7
The disjunction of the propositions p and q is the
proposition
p q : 5 < 9 or 9 < 7
Example 1.4
Consider the following propositions
p : It is Friday
q : It is raining
Construct the propositions p
q and p
q
The conjunction of the propositions p and q is the
proposition
p q : It is Friday and it is raining
The disjunction of the propositions p and q is the
proposition
p q : It is Friday or It is raining
8/11/2019 Discrete Mathematics - Lecture 1 - Propositions
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8/11/2019 Discrete Mathematics - Lecture 1 - Propositions
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Lecture 1: Propositions and Compound Propositions P a g e | 3
EQUIVALENT PROPOSITIONS
Two propositions are equivalent if they have exactly
the same truth values under all circumstances. We
write p q.
Example 1.9
a.
Show that (p q) p q.
Note: a and b are known as DeMorgan's laws.
Solution:
p q p q (p q) p q p q
F F F T T T T
F T T F T F F
T F T F F T F
T T T F F F F
Assignment / Quiz
b. Show that (p q) p q:
c. Show that (p) p
d. Show that p q q p and p q q p
e. Show that (p q) r p (q r)
f. Show that (p q) r p (q r)
g.
Show that (p q) r (p r) (q r)
h. Show that (p q) r (p r) (q r)
i. Show that (p q) pq
CONTRADICTION
A compound proposition that has the value F for all
possible values of the propositions in it is called a
contradiction.
Example 1.10
Show that the proposition pp is a contradiction.
Solution:
p p p p
F T F
T F F