- a declarative sentence that is either true or
false but not both.
- usually represented by letters such as p, q, r, and s
which are known as propositional variables
Examples:
p:
q:
r:
s:
Today is Sunday.
Crocodiles are smaller than alligators.
2+2=5
-5 > -10
The following are NOT propositions:
1.
2.
3.
4.
Go home.
(a command)
Are you a Catholic? (a question)
x+1=4
(the value of x is not given)
What a beautiful day! (an exclamation)
NEGATION (denoted by  or  )
p or p
(read as “not p”)
The negation of p is “not p”
Examples:
p: Today is Monday.
p: Today is NOT Monday.
r: He has a cellphone.
r: He does not have a cellphone.
p: The virus spreads quickly.
p: The virus does not spread quickly.
r: The students attend the meeting.
r: The students do not attend the meeting.
q: y  5
q: y < 5
It
is false that . . .
It is not true that . . .
It is not the case that . . .
The Connectives
1. Conjunction – denoted by 
Ex: p  q (read as “p and q”)
2. Disjunction – denoted by 
Ex: p  q (read as “p or q”)
3. Conditional – denoted by →
Ex: p → q (read as “if p then q”)
4. Biconditional) – denoted by 
Ex: p  q (read as “p if and only if q”)
Let
p: He is respectful.
q: He studies at UNO-R.
Write the following in symbolic form.
1. He studies at UNO-R and he is respectful.
Answer: q  p
2. He is not respectful. Answer: p
3. If he is not respectful, then he does not study
at UNO-R. Answer:
4. He studies at UNO-R if and only if he is
respectful. Answer:
5. He is respectful or he does not study at UNO-R.
Answer:
Let
p: Today is Saturday.
q: I will watch a movie.
Write the following in statement form:
1. p → q
Ans: If today is Saturday, then I
will not watch a movie.
2. q  p
Ans:
3. q  p
Ans:
Parentheses are used in the following cases:
1. The use of “It is false that…” and “It is not true that…”
negates everything that follows.
It is false that p and q.
 (p  q)
2. Statements on the same side of a comma are
grouped together.
p, and q or not r
p  (q   r)
p and q, or r
(p  q)  r
If p and not q, then r or s
(p  q) → (r  s)
3. “Neither p nor q” is the same as “not p or q”
(p q)
1. Paolo went to school (S) and took the test (T), or he is with
his friends (F).
Ans: (S  T)  F
2. It is not true that it rained last night (R) or there was a
brownout (B).
Ans:
3. Lebron James is a basketball player (B), if and only if he is
not a football player (F) and he is not a rock star (R).
Ans:
4. It is false that Philippines is hopeless (H) and martial law is
exercised (M).
Ans:
5. If you take a vacation (V), then you will neither attend the
seminar (S) nor go to picnic with us (P). Ans:
 (p  q)  p  q
 (p  q)  p  q
meaning, the negation of
➢ “not p and q” is equivalent to “not p or not q”
➢ “not p or q” is equivalent to “not p and not q”
Example:
Write the negation of the statement:
1. She is cute and she has dimples.
Answer: She is not cute or she does not
have dimples.
2. Covid spreads quickly or the people
wear masks.
Answer:
A. Universal Quantifiers
1. all, every: assert that every element of
a given set satisfies some conditions
2. none, no: denotes the none-existence
of something
B. Existential Quantifiers – used to assert the
existence of something
some, there exists, at least one
Statement
Negation
All x are y
Some x are not y
No x are y
Some x are y
Write the negation of:
1. Some airports are open.
Answer: No airports are open.
2. All cellphones are expensive.
Answer:
3. No odd numbers are divisible by 2.
Answer: