Propositions

Consider the following

statements:
One plus one equals three.
Today is Sunday.
Good morning, class!
Get out of my sight!
Simple Propositions
Simple Propositions – are simple
statements that can be decided
whether true or false.
*Propositions are usually denoted by small
letters.
Examples:
p: Mindanao is an island in the
Philippines.

q: 3 + 2 = 5.
 Compound Propositions
Compound Propositions – are composite
statements that are composed of sub-
statements and various logical connectives.

*These are formed from simple

propositions using logical connectors.
Some logical connectors involving
propositions p and/or q may be
expressed as follows:
not p
p and q
p or q
If p, then q
Examples:
p: You study hard
q: You get good grades
If you study hard, then you will
get good grades.
Examples:
p: You are more than 60 years old
q: You are entitled to a Senior
Citizen’s card
If you are more than 60 years old, then
you are entitled to a Senior Citizen’s
card.
Tell whether each given
statement is a proposition or not.

1.) Laguna is in Manila. Proposition

*this dictates a false statement

2.) 2 + 2 = 4. Proposition
*this dictates a true statement
Tell whether each given
statement is a proposition or not.

 Where are you going?

Not a proposition
*this can be neither true nor false
 Putyour homework on the blackboard.
Not a proposition
*this can be neither true nor false
 Determine whether each of the following
statements is a proposition or not. If it is a
proposition, tell if it is simple or compound.
Not a proposition 1. Mabuhay!
Simple proposition 2. Jose Rizal is our National Hero.
Compound proposition 3. Our logic teacher is either pretty
or handsome.
Compound proposition 4. 2 is even and prime.
Compound proposition 5. If an integer is even, then its
square is also even
Logical Connectives - are used to
symbolize compound statements.

Connectives Symbol
not  not p, not q
and, but Ʌ p and q/p but q
or V p or q
Examples: Using p and q, write each of the
following statements in symbols.
p: The sun is shining.
q: The sky is blue.
1. The sun is shining, and the sky is
blue.
In symbols: p Ʌ q
2. The sun is shining, but the sky is not blue.
In symbols: p Ʌ q
3. It is false that the sun is shining or the
sky is blue.
In symbols:  (p V q)
4. Neither the sun is shining nor the sky is
blue.
In symbols:  p V  q
Operations and
Truth Values of
Propositions
Truth table - it shows the
truthfulness or falsity of a
statement.
Truth value of a proposition is
known once the truth value of
each of its variables is known.
Negation - the truth value of the
negation of a statement is always
the opposite of the value of the
original statement.
p p
T F
F T
Conjunction - the conjunction
of two statements is true only
if each component is true.
p q pɅq
T T T
T F F
F T F
F F F
Disjunction - the disjunction
of two statements is false only
if each component is false.
p q pVq
T T T
T F T
F T T
F F F
Example: Construct the truth
table for the proposition p Ʌ q.
p q q p Ʌ q
T T F F
T F T T
F T F F
F F T F
Construct the truth table for the
proposition q V (p Ʌ q).
p q pɅq q V (p Ʌ q)
T T T T
T F F F
F T F T
F F F F
Construct the truth table for the
proposition (p Ʌ q) V p.
p q pɅq (p Ʌ q) V p
T T T T
T F F T
F T F F
F F F F
Construct a truth table for
p V [p Ʌ(q V q)].
p p q q qV q p Ʌ (q V q) pV[p Ʌ(q V q)]
T T
T F
F T
F F
 Construct a truth table for
p V [p Ʌ(q V q)].

p p q q q V q p Ʌ (q V q) p V [p Ʌ (q V q)]

T F T F T F T
T F F T T F T
F T T F T T T
F T F T T T T
Conditional
Statements
Consider the following statements:

1. If you get good grades, then you will get

into a good college.

2. A triangle is isosceles if and only if it has

two congruent sides.
Conditional Statement
A conditional statement is a statement that can be
written in if-then form. “If p then q.”
Conditionals can also be read as:
- p implies q
- p only if q
- p is sufficient for q
- q is necessary for p

Example: If your feet smell and your nose runs, then you're
built upside down.
To illustrate conditional statements, the symbol ‘→’ is used.
It is true except in the case where p is true and q is false.
Example: If you get good grades, then you will get into a good
college. In symbols: p → q.

p q p→q
T T T
T F F
F T T
F F T
Biconditional statements are propositions in
the form ‘p if and only if q’.

To illustrate biconditional statements, the symbol

‘ ’ is used.
It is true in the cases where p and q are both
true or p and q are both false.
Example: A triangle is isosceles if and only if it
has two congruent sides. In symbols: p q.

p q p q
T T T
T F F
F T F
F F T
 Determine whether each given statement is
conditional or biconditional.

Conditional 1. If 1 + 1 = 3, then 1 + 2 = 3.
Biconditional 2. 1 + 1 = 3 if and only if Mars is a black
hole.
Biconditional 3. A polygon is a triangle if and only if it
has exactly three sides.
Conditional 4. If the earth is round, then Mars is flat.
Conditional 5. If there is life on Mars, then we should
fund NASA.
Logical Equivalence
Two propositions are said to be logically equivalent if their truth tables
are identical.

Example: Show that (p→q)Ʌ(q→p) and p q are logically

equivalent.

p q p→q q→p (p→q)Ʌ(q→p) p q p q

T T T T T T T T
T F F T F T F F
F T T F F F T F
F F T T T F F T
Logical Equivalence
p q p→q q→p (p→q)Ʌ(q→p) p q p q
T T T T T T T T
T F F T F T F F
F T T F F F T F
F F T T T F F T

Since the truth value of (p→q)Ʌ(q→p) and p q are

equal, then the given propositions are logically
equivalent or (p→q)Ʌ(q→p)  p q.
Do the following:
Complete the following truth tables.
A. p→(rɅq)
p q r rɅq p→(rɅq)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
B. p (rVq)

p q r rVq p (rVq)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
C. (pɅq) (r→p)

p q r p pɅq r→p (pɅq) (r→p)

T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F