LESSON 9 & 10 - LOGIC STATEMENTS, CONNCETIVES, QUANTIFIERS, AND TRUTH TABLE..pptx

TRECE MARTIRES CITY
COLLEGE
INSTRUCTOR:
QUARRE, FRANCIS R.
Bachelor of Secondary Education Major in
Mathematics
LOGICAL STATEMENTS
and TRUTH TABLE

(Statements, Connectives,
and Quantifiers)
LOGICAL STATEMENTS

STATEMENT
A statement in logic is a declarative
sentence that is either true or false. We
represent statements by lowercase letters
such as p, q, or r.

EXAMPLES of STATEMENTS:
1. Guillermo has the flu.
2. Today is Monday.
3. If Paul eats pizza, then Paul will drink
soda.

EXAMPLES of NON-STATEMENTS:
1. Go clean your room.
2. Why are you here?
3. This sentence is false.
(It cannot be either true or false. If we assume that this
sentence is true, then we must conclude that it is false. If we
assume that it is false, then we must conclude that it is true.)
(Interrogative)
(Command)
(Paradox)

CONNECTIVES
 NEGATION
 expresses the word “NOT” and uses
the symbol : not p is notated
A statement p and its negation ~p will always have
opposite truth values.

EXAMPLE
: Manila is the capital of the Philippines.
: Manila is not the capital of the Philippines.

He’s always late.
He’s not always late.
EXAMPLE

She sometimes apologizes.
She never apologizes.
EXAMPLE

I am not going to the basketball game.
I am going to the basketball game.
EXAMPLE

CONNECTIVES
 CONJUNCTION
 expresses the word “AND” between
two statements and uses the symbol
∧: p and q is notated p q
∧
In general, in order for any statement of the form
“p q” to be true, both p and q must be true.
∧

EXAMPLE
p: It is raining.
q: The sun is shining.
The conjunction of p and q can be represented
as, pΛq: It is raining and the sun is shining.

EXAMPLE
p: The sky is blue.
q: The grass is green.
as, pΛq: The sky is blue and the grass is green.

EXAMPLE
p: The sun is a star.
q: The earth is flat.
as, pΛq: The sun is a star and the earth is flat.

CONNECTIVES
 DISJUNCTION
 expresses the word “OR” between two
statements and uses the symbol ∨:
p or q is notated p q
∨
In general, in order for a statement of the form p q to be true, at
∨
least one of its two parts must be true. The only time a disjunction is
false is when both parts (both“components”) are false.

EXAMPLE
p:The measure of A is greater than B.
∠ ∠
q: The measures of the angles ( ’s) A and B
∠
are equal.
The disjunction of p and q can be represented
as, p q
∨ : The measure of A is greater than
∠
B,
∠ or they are equal.

EXAMPLE
p: I will go to the store.
q: I will stay at home.
as, p q
∨ : I will go to the store or I will stay at
home.

EXAMPLE
p: The sky is purple.
q: The grass is red.
as, p q
∨ : The sky is purple or the grass is red.

EXAMPLE
p: 2 + 2 = 4
q: 1 + 3 = 4
as, p q
∨ : 2 + 2 = 4 or 1 + 3 = 4

CONNECTIVES
 CONDITIONAL
 statement is of the form “If ... then ...”
and uses the symbol → : If p, then q is
notated p→q
Conditional statement is considered false only if the hypothesis is
true and the conclusion is false. In all other cases it is considered
true.

EXAMPLE
p: I eat breakfast.
q: I feel good all day.
The conditional of p and q can be represented
as, p→q: If I eat breakfast then I feel good all
day.

EXAMPLE
p: It is raining.
q: The ground is wet.
as, p→q: If it is raining then the ground is wet.

EXAMPLE
p: It is not raining.
q: The ground is not wet.
as, p→q: If it is not raining then the ground is
not wet.

EXAMPLE
p: It is not raining.
q: The ground is wet.
as, p→q: If it is not raining then the ground is
wet.

CONNECTIVES
 BICONDITIONAL
 statement expresses “if and only if”
between two statements and uses the
symbol ↔ : p if and only if q is notated
p↔q
Biconditional statement is true when both components have the
same truth value (either both true or both false), and false
otherwise.

EXAMPLE
p: Four sides of a rectangle are equal.
q: It is a square.
The biconditional of p and q can be represented
as, p↔q: A rectangle is a square if and only if
its four sides are equal.

EXAMPLE
p: An angle measures 90 degrees.
q: It is a right angle.
as, p↔q: An angle is right if and only if it
measures 90 degrees.

EXAMPLE
p: Two angles are congruent.
q: They have the same measure.
as, p↔q: Two angles have the same measure if
and only if they are congruent.

Compound
Statements in
Symbolic
Form

COMPOUND STATEMENTS
Compound Statement
 is a statement that is formed by combining two
or more statements using special words called
connectives.

LESSON 9 & 10 - LOGIC STATEMENTS, CONNCETIVES, QUANTIFIERS, AND TRUTH TABLE..pptx

Quantifiers
are another type of phrase or
a special word used in
mathematical statements.
it refers to quantities that the
number of elements in the
domain which satisfy the
particular simple statement.

Existential quantifier states that a set contains
at least one element. (some, many, or at least
one)
Universal quantifier states that an entire set
of things share a characteristic. (all, every, or
none)

QUANTIFIED STATEMENTS
1. All dogs are poodles.
2. Some books have hard covers.
3. No Philippine president is from Mindanao.
4. Some cats are mammals.
5. Some cats aren't mammals.
According to your everyday experience, decide whether
each statement is true or false.
(because there is at least one cat that is a mammal; in
fact every cat is a mammal).
T
F
F
T
F

Negating a Quantified
Statement
When we negate a statement
with a universal quantifier, we
get a statement with an
existential quantifier, and vice-
versa.

 The negation of “all A are B” is “at least one A is not B”.
 The negation of “no A are B” is “at least one A is B”.
 The negation of “at least one A is B” is “no A are B”.
 The negation of “at least one A is not B” is “all A are B”.
Negating a Quantified Statement

Truth Table
 a table showing what the
resulting truth value of a
complex statement is for all the
possible truth values for the
simple statements.
 a device that allows us to
analyze and compare
compound logic statements.

Rules for Constructing a Truth Table
The number of rows that a truth table needs is
determined by the number of basic statement
letters involved in the set of formula.
1 variable---2 rows
2 variables--4 rows
3 variables--8 rows

TRUTH TABLE:
As an introduction, we will make truth
tables for the following statements:
1.
2. ∧
3. ∨
4. ~
∨

TRUTH TABLE: As an introduction, we will make truth tables
for these two statements: 1.
Note: In negation, it will always have opposite truth
values.
T
F

for these two statements: 1.
Note: In negation, it will always have opposite truth
values.
T F
F T

for these two statements: 2. p q
∧
∧
T T
T F
F T
F F
Note: In conjunction, both statements needs to be true to
conclude that it is TRUE.

∧
∧
T T T
T F F
F T F
F F F
Note: In conjunction, both statements needs to be true to
conclude that it is TRUE.

∨
∨
T T
T F
F T
F F
Note: In disjunction, at least one of its two parts must be true.
The only time a disjunction is false is when both parts
(both“components”) are false.

∨
∨
T T T
T F T
F T T
F F F
Note: In disjunction, at least one of its two parts must be true.
The only time a disjunction is false is when both parts
(both“components”) are false.

for these two statements: 4. ~
∨
~ ∨~
T T
T F
F T
F F

∨
~ ∨~
T T F
T F T
F T F
F F T

∨
~ ∨~
T T F T
T F T T
F T F F
F F T T

EXERCISE :
Construct a truth table given the
following statements:
1. ∨
2. ~
∧
3. ~
∧ ∨

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