General Mathematics. Quarter 2 Module 5.

A proposition is a declarative
sentence that is either true or
false, but not both.

 If a proposition is true,
then its truth value is
TRUE which is denoted by
T; otherwise, its truth value
is FALSE and is denoted
by F.

Example:
1.Two plus two equals four.
- True statement
2.Two plus two equals five.
- False statement

 Propositions are usually denoted
by letters (p, q, r, s)
Example:
P – Everyone should study logic.
may be read as:
P is the proposition “Everyone
should study logic.”

 Some sentences are not proposition
•Questions
•Exclamations
•Commands
•Requests
•Stipulations (rules)
•Wishes
•Nonsense

Example:
1. Aren’t you coming then?
2. Please place your hand on your
head if you want to go home.

Determine whether each of the
following statements are proposition or
not. If a proposition, give its truth value.
P: Mindanao is an island of the Philippines.
- It is a declarative sentence, and
Mindanao is an island in the
Philippines.
- P is a true proposition.

Q: Find a number which divides your
age.
- This is an imperative sentence.
- It is not a proposition.
R: Welcome to the Philippines!
- Statement R is an exclamatory
sentence.

S: What is the domain of the
function?
- It is an interrogative sentence.

A compound proposition is a
proposition formed from simpler
proposition using logical
connectors or some combination
of logical connectors.

Some logical connectors involving
propositions p and/or q may be expressed
as follows:

1.What is a proposition?
 A proposition is a declarative
sentence that is either true or
false, but not both.

2. What are the classifications of
propositions?
 Simple Proposition
 Compound proposition

3. How to distinguish the difference between simple
and compound propositions?
 A compound proposition is a proposition
formed from simpler proposition using logical
connectors or some combination of logical
connectors.
 A proposition is simple if it cannot be broken
down any further into other component
propositions.

Objectives:
K: identify the definition of logical operators used
in a proposition
S: perform the different types of operations on
propositions
A: develops perseverance and patience in
performing the different operations of propositions
ESTABLISHING A PURPOSE FOR THE LESSON

Given:
Let p: He has green thumb. q: He is a senior citizen.
Convert each compound propositions into symbol.
____1) He has green thumb, and he is a senior citizen.
____2) He does not have a green thumb, or he is not a
senior citizen.
____3) It is not the case that he has green thumb or is a
senior citizen.
____4) If he has green thumb then he is not a senior
citizen.
PRESENTING EXAMPLES /INSTANCES OF THE NEW LESSON
(̴ p) ˅ ( ̴ q)
(̴ p) ˅ q
(p → ( ̴ q)
p ˄ q

DISCUSSING NEW CONCEPTS AND PRACTICING NEW SKILLS

Conjunction – two simple propositions connected
using the word and/but.
Example:
 5 is a factor and multiple of 25.
 Today is Tuesday and tomorrow is Wednesday.
 Torta is a type of traditional bread, and it is the
famous bread in Zamboanguita.

Disjunction – two simple propositions connected
using the word or.
Examples:
 I will pass the Math exam, or I will be promoted.
 Either a student takes Mathematics elective next
semester, or he takes a business elective next
year.
 Neither Mathematics is difficult, nor PE is easy.

Conditional – two simple propositions that are
connected using the words if…then.
Example:
 If Mathematics is difficult, then PE is easy.
 You are entitled to a php 30, 000 discount if you
are a member.
 If Vance is in grade 11, then she is a senior high
school student.

Biconditional – two propositions that are
connected using the words if and only if.
Example:
 PE is easy if and only if Mathematics is difficult.
 Two sides of a triangle are congruent if and only
if two angles opposite them are congruent.
 The crime rates will go up if and only if there is
an unemployment problem.

Negation – a statement that is false whenever
the given is true and true whenever the given
statement is false.
The negation can be obtained by inserting the
word not in the given statement or by prefixing
it with phrases such as “It is not the case
that…”

Example:
 p: I like coffee.
Negation: “I don’t like coffee” or “It is not
the case that I like coffee”
 r: February 14 is a holiday.
Negation: It is false that February 14 is a
holiday.

Direction: Symbolize the following propositions.
H: I have missed the bus. S: I slept in.
L: I am late. B: The bus is late.
W: I will get to work on time.
1. The bus is not late.
2. I am late, and I have missed the bus.
3. The bus is not late but I am.
4. It’s not true that the bus late and I am late.
5. If the bus is not late then I have missed it.
6. I am not late, and I did not sleep in.
7. Either the bus is late or I am.
8. If I slept in, then I’m late and I have missed the bus.
9. I won’t get work on time.
10. If the bus is late or I am, it won’t get to work on time.

Negation
The negation is the simplest operation
of propositions. This is that operation
(function) of proposition p which is true
when p is false, and false when p is
true.

Formally, If p is a proposition variable,
the negation of p is "not p" or "It is not
the case that p." Symbolically, we shall
denote the general negation of p by ~p
or ¬p. The prefixed symbol is called a
tilde. Negation of p has opposite truth
value form p. That is, if p is true, then ~p
is false; if p is false, ~p is true.

More commonly, we use internal
negation to negate a sentential
(proposition variables.) For instance,
"b is a rational number" becomes "b is
not a rational number," and "x = 5"
becomes "x ≠ 5", and so on.
If p is true, then ~p is false; and if p is
false, then ~ p is true.

The truth value of ~p
may be defined
equivalently by the
table below. Thus the
truth value of the
negation of p is
always the opposite
of the truth of p.

Example 1. State the negation of the
following propositions.
p: 2 is an odd number.
q: Everyone in the Visayas speaks
Cebuano.
Solution:
~p: It is not true that 2 is an odd number.
~q: Not everyone in Visayas speaks
Cebuano.

Conjunction
The joining of two or more propositions by the
word "and" results in their so-called
conjunction or logical product; the
propositions joined in this manner are called
the members of the conjunction or the
factors of the logical product. The
conjunction, "p and q", has truth for its truth-
value when p and q are both true; Otherwise
it has falsehood for its truth-value.

Formally, If p and q are proposition
variables, the conjunction of p and q is a
compound proposition "p and q." We
symbolize the logical conjunction of p and
q by p q. It is true when, and only when,
both p and q are true. If either p or q is
false, or if both are false, p q is false.
Equivalently,
If p and q are true, then p q is true;
otherwise p q is false.

The truth values
for conjunction
(or equivalently,
the truth table for
the operator )
are depicted in a
following table.

Disjunction
The joining of two or more propositions by
the word "or" results in their so-called
disjunction or logical sum; the propositions
joined in this manner are called the
members of the disjunction or the summands
of the logical sum. The disjunction, "p or q",
has truth for its truth-value when p is true and
also when q is true, but if falsehood when
both p and q are false.

Formally, If p and q are proposition
variables, the disjunction of p and q is "p
or q" and we symbolize the logical
disjunction of p or q by p q. It is true
when at least one of p or q is true and is
false only when both p and q are false.
Equivalently,
If p and q are false, then p q is false;
otherwise p q is true.

The truth values for
disjunction (or
equivalently, the
truth table for the
operator ) are
depicted in a
following table.

In logic, we avoid possible ambiguity
about the meaning of the word "or" by
understanding it to mean the inclusive
"and/or." In logic, we generally used "or" in
the nonexclusive sense. For instance, to
say "x is rational or it is negative" does not
mean that it cannot be both rational and
negative. We shall use consistently the
interpretation that at least one of the
choices must hold, and both may be hold.

Conditional
We shall talk implication in a separate section, in
depth. Here, we simply define and talk about the
meaning in a general sense. We interpret the
meaning of Implication, i.e. "p implies q," or "if p, then
q" as "Unless p is false, q is true," or "either p is false or
q is true," where p and q are proposition variables.
That is to say, "p implies q" is to mean "~p or q"; its
truth-value is to be truth if p is false, likewise if q is true,
and is to be falsehood if p is true and q is false. The
fact that "implies" is capable of other meanings does
not concern us: For the time being this is our
"meaning" and we are sticking to it.

Accordingly, the
truth table for the
"implies" or "p →
q" is shown
below:

Biconditional
The biconditional of propositions
p and q is denoted by
p ↔ q: (p if and only if q)
and is defined through its truth
table:

The proposition
may also be written
as ‘p iff q’. the
propositions q and
q are the
components of the
biconditional.

OPERATIONS ON PROPOSITIONS
1. p r
2. ( r) q
3. p→r
4. ( q) ( s)
5. p (r s)
Example:
Find the truth values of the following propositions using
operations on proposition for the given scenario.

CONSTRUCTING TRUTH TABLE
A truth table shows all possible truth
value combinations of two or more given
propositions. The number of rows of the
table can be computed using the formula
, where is the number of
propositions.

Let p and q be two propositions, to
identify the number of rows, we
have , is 4. Therefore, there
are 4 rows in the truth table.
However, if there are 3
propositions, then , is 8.
Therefore, there are 8 rows in the
truth table.

Example 1: Let p and q be propositions.
Construct the truth table of the following
compound proposition
1.a. (p˅q)˄p 1.b. (p→q)˄(p˅q)

*Remember: In constructing truth table, start with the column
with the simple compound proposition then to the complex
compound proposition.
Example 2: Express the following
propositions in symbols, where p, q, r and s
are defined as follows, and find its truth value.
p: John is a big eater.
q: Carl plays guitar.
r: Mark likes to travel.
s: Ces is a performer.

Example 2.a: John is a big eater and Mark likes
to travel.
John is a big eater and Mark likes to travel.

Example 2.b: It is not true that Carl plays guitar
and Ces is a performer.
It is not true that Carl plays guitar and
Ces is a performer.

Example 2.c: It is not true that Carl plays guitar
and Ces is a performer.
If John is a big eater or Carl plays guitar,
→ p v q
then Mark likes to travel.
r

A proposition that is always true
is called a tautology (T), while a
proposition that is always false is
called a contradiction (C).

ASSIGNMENT: Construct the
Truth table of values of the
following Compound propositions.
1.{[(P ^ Q) ^ (Q ^ P)] ^ P} (P v Q)
2.[(P Q) ^ P] Q
3.[(P Q) ^ (Q R)] ( R P)
4.[(P Q) ^ P] Q

General Mathematics. Quarter 2 Module 5.

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