DEFINITION 3. Disjunction of p and q
LESSON 6: TRUTH TABLE
The truth value of a propositional form can be
shown through a truth table. If a propositional form
has n propositional variables as components, then
its corresponding truth value has 2n number of
rows.
DEFINITION 1. Negation of p
Let p be a proposition.
The statement “It is not the case that p” is also a
proposition, called the “negation of p” or ¬p (read
“not p”)
p = The sky is blue.
¬p = It is not the case that the sky is blue.
¬p = The sky is not blue.
NOTE:
T=F
F=T
Let p and q be propositions.
The proposition “p or q,” denoted by pVq, is the
proposition that is false when p and q are both
false and true otherwise.
NOTE: (T DOMINANT)
T+T=T
If not the same like (T + F) or (F + T) always = T
F+F=F
Table 3. The Truth Table for the
Disjunction of two propositions
p
q
pVq
T
T
F
F
T
F
T
F
T
T
T
F
DEFINITION 5. Implication p → q
Table 1. The Truth Table for the
Negation of a Proposition
p
¬p
T
F
F
T
DEFINITION 2. Conjunction of p and q
Let p and q be propositions.
The proposition “p and q,” denoted by p∧q is true
when both p and q are true and is false otherwise.
This is called the conjunction of p and q.
NOTE:
T+T=T
F+F=F
If not the same = F
Table 2. The Truth Table for the
Conjunction of two propositions
p
q
p∧ q
T
T
F
F
T
F
T
F
T
F
F
F
Let p and q be propositions.
The implication p → q is the proposition that is
false when p is true and q is false, and true
otherwise. In this implication p is called the
hypothesis (or antecedent or premise) and q is
called the conclusion (or consequence).
NOTE
If T is in the right side the answer is = T
If F is in the right side the answer is = F
F+F=T
T+T=T
Table 5. The Truth Table for the
Implication of p→q.
p
q
p→q
T
T
T
T
F
F
F
T
T
F
F
T
●
If p, then q
●
p is sufficient for q
●
p implies q
●
q if p
●
if p,q
●
q whenever p
●
p only if q
●
q is necessary for p
●
Not the same as the if then construct used in
programming languages such as If p then S
How can both p and q be false, and p→q be true?
- Think of p as a “contract” and q as its
“obligation” that is only carried out if the
contract is valid.
Example:
“If you make more than P300,000 annually, then you
must file a tax return.” This says nothing about
someone who makes less than P300,000 annually.
So, the implication is true no matter what someone
making less than P300,000 does.
p: Bill Gates is poor.
q: Pigs can fly.
p→q, 0→0
p→q is always true because Bill Gates is not poor.
Another way of saying the implication is “Pigs can
fly whenever Bill Gates is poor” which is true since
neither p nor q is true.
Converse of p → q is q → p
Contrapositive of p → q is the proposition ¬q → ¬p
DEFINITION 6. Biconditional
Let p and q be propositions.
The biconditional p ↔ q is the proposition that is
true when p and q have the same truth values and
is false otherwise. “p if and only if q, p is necessary
and sufficient for q”
Table 6. The Truth Table for the biconditional p ↔ q
p
q
p↔q
T
T
F
F
T
F
T
F
T
F
F
T
NOTE:
T+T=T
F+F=T
If there's an F in “any side” it is = F
IMPORTANT:
Number of Variable(letters/numbers = exponent or
number of column
● 2n
Example:
p ↔ q = 2 Variable = 22 = 4 columns
r V (p ↔ q) = 3 Variable 23 = 8 columns