The document outlines concepts related to logical statements in mathematics, including the distinction between simple and compound statements, and introduces logical connectives and truth values. It explains various forms of conditional and biconditional statements, providing examples and truth tables to illustrate the concepts. Additionally, it covers the relationships between different types of statements, such as converse, inverse, and contrapositive.

1. ERWIN A. MARIANO,MAED-MATH
erwinmariano27@gmail.com
MATHEMATICS IN THE
MODERN WORLD
GEC 004

2. 1. Differentiate simple and compound logical
statement.
2. Familiarize with the logical connectives and
symbols
3. Give the truth values using truth tables
GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
OBJECTIVES

3. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
A statement (or proposition) is a
declarative sentence which is either true
or false, but not both.
Determine whether each sentence is a
statement.
a. Manila is in the Philippines.
b. How are you?
c. 3 is a prime number
d. x + 4 = 15

4. Simple Statement- is a statement that conveys a
single idea.
GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
Compound Statement- is a statement that conveys
two or more ideas
p: Today is Sunday.
r: Prof. Mariano is handsome.
q: I love Mathematics in the Modern World.
s: I am going to the basketball game.

5. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
Type of
Statement
Connective Statement
Symbolic
Form
Negation not not p ~ 𝑝
Conjunction And p and q 𝑝 ∧ 𝑞
Disjunction Or p or q 𝑝 ∨ 𝑞
Conditional If… then If p, then q. 𝑝 → 𝑞
Biconditional If and only if
p if and only if
q
𝑝 ↔ 𝑞
Truth Table- shows the truth value of a compound statement for all possible
truth value of its simple statements.

6. Consider the following statement:
GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
1. Today is Sunday and Prof. Mariano is handsome.
Write the following compound statements in
symbolic form.
p: Today is Sunday.
r: Prof. Mariano is handsome.
q: I love Mathematics in the Modern World.
s: I am going to the basketball game.
2. I love MMW or I am not going to the basketball
game.
3. If today is Sunday, then I am not going to the
basketball game.

7. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
p: Today is Sunday.
r: Prof. Mariano is handsome.
q: I love Mathematics in the Modern World.
s: I am going to the basketball game.
5. If Prof. Mariano is not handsome, then I don’t love
Mathematics in the Modern World.
6. Today is Sunday if and only if Prof.Mariano is
handsome.
7. I’m not going to the basketball game if and only if
today is not Sunday.

8. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENT
p: Today is Sunday.
r: It is raining.
q: I will take a bath.
s: I will not brush my teeth.
1. p ∧ 𝑠
2. r ∧∼ 𝑞
3. q ∨ s
4. 𝑝 → 𝑠
5. ∼ 𝑝 →∼q
6. 𝑝 ↔ ~𝑠
7. ~𝑟 → 𝑞
8. 𝑝 ↔ 𝑠
9. ~𝑟 → 𝑞 ∨ 𝑠
10.(p ∧ 𝑠) ↔ (~𝑟 ∨ 𝑞)
Given the following
statement.
Translate each of the
following symbols

9. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
CONJUNCTION
The conjunction of the statement p and q is the compound
statement “p and q.”
Symbolically, p  q, where  is the symbol for “and.”
Property 1: If p is true and q is true, then p  q is true;
otherwise p  q is false. Meaning, the conjunction of
two statements is true only if each statement is true.
p q p  q
T
T
F
F
T
F
T
F
T
F
F
F

10. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
DISJUNCTION
The disjunction of the statement p, q is the compound
statement “p or q.”
Symbolically, p  q, where  is the symbol for “or.”
Property 2: If p is true or q is true or if both p and q are true,
then p  q is true; otherwise p  q is false. Meaning,
the disjunction of two statements is false only if
each statement is false.
p q p  q
T
T
F
F
T
F
T
F
T
T
T
F

11. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
CONDITIONAL STATEMENT
The conditional (or implication) of the statement p and q is the
compound statement “if p then q.”
Symbolically, p  q, where  is the symbol for “if then.” p is called
hypothesis (or antecedent or premise) and q is called conclusion (or
consequent or consequence).
Property 4: The conditional statement
p  q is false only when p is true and q is
false; otherwise p  q is true. Meaning
p  q states that a true statement cannot
imply a false statement.
p q p  q
T
T
F
F
T
F
T
F
T
F
T
T

12. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
CONDITIONAL STATEMENT
If p, then q. Every p is a q.
If p, q. q, if p
p only if q. q provided that p.
p implies q. q is a necessary condition for p.
Not p or q. p is a sufficient condition for q.
Common Forms of p → q
Every square is a rectangle.
If a figure is a square, then it is a rectangle.

13. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENTS RELATED TO THE CONDITIONAL STATEMENT
Converse
Inverse
Contrapositive
1. The Converse of p → q is formed by interchanging the
antecedent p and the consequent q.
2. The Inverse of p → q is formed by negating the antecedent
p and negating the consequent q.
3. The Contrapositive of p → q is formed by negating both
the antecedent p and the consequent q and interchanging
these negated statements.

14. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENTS RELATED TO THE CONDITIONAL STATEMENT
CONDITIONAL CONVERSE INVERSE CONTRAPOSITIVE
p q p → q q → p ∼p →∼ q ∼q →∼ p
T T T T T T
T F F T T F
F T T F F T
F F T T T T

15. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
statements
-The Conditional
and Related
Statements
STATEMENTS RELATED TO THE CONDITIONAL STATEMENT
Every square is a rectangle.
Conditional:
Converse:
Inverse:
Contrapositive:
If a figure is a square, then it is
rectangle
If a figure is a rectangle, then it
is a square.
If a figure is a not a square, then
it is not a rectangle.
If a figure is a not rectangle, then
it is not a square.

16. GEC 004 MATHEMATICS IN THE MODERN
SECTION 2.
MATHEMATICS
AS A TOOL
LOGIC
-The conditional
and Biconditional
Statements
-The Conditional
and Related
Statements
STATEMENTS RELATED TO THE CONDITIONAL STATEMENT
The sum of the measures of two
complementary angles is 90 degrees.
Conditional:
Converse:
Inverse:
Contrapositive:
If two angles are complementary, then the
sum of the measures is 90 degrees.
If the sum of the measures of two angles is
90 degrees, then they are complementary
If two angles are not complementary, then
the sum of the measures is not 90 degrees
If the sum of the measures of two angles is not 90
degrees, then they are not complementary

17. GEC 004 MATHEMATICS IN THE MODERN