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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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