This document provides an overview of propositions and logic. It discusses: - Propositions as statements that can be classified as true or false. - Simple and compound propositions, where compound propositions combine two or more simple propositions. - Logical operations on propositions including negation, conjunction, disjunction, conditional, and biconditional statements. - Truth tables as a way to systematically determine the truth value of compound propositions based on the truth values of their components.

1. MATHEMATICS IN THE
MODERN WORLD
BOARD
Lesson7:
PROPOSITIONS

2. BOARD
 the study of the methods and
principles used to distinguish good
(or correct) from bad (or incorrect)
reasoning
 focuses on the relationship among
statements rather than to the content
of just one statement.

3. Topic 1: LOGIC AND PROOFS
BOARD
An argument is a set of statements, one
of which is called the conclusion and the
rest of which are called premises. An
argument is said to be valid if the
conclusion must be true whenever the
premises are all true. An argument
is invalid if for all the premises to be true
and the conclusion to be false.
Consider the following
two arguments:
If Diane eats her
vegetables, then she can
have a cookies.
Diane eats her vegetables.
∴ Diane gets a cookies
(The symbol “∴” means
“therefore”)

4. BOARD
 a declarative sentence that can be classified
as true or false, but not both
Proposition
 a self-contradictory proposition
Paradox
1. Rowena is passing in Mathematics.
2. Osaka is the capital of Japan.
3. 5+3=8 and 12-7=5
4. The number 4 is even and less
than 12.
5. 2+2=3
6. Save money by spending it.
 sentence that cannot be classified as either
true or false
Topic 1: Simple and Compound Proposition
A true proposition has a truth value “True”
and a false proposition has truth value “False”.
The symbols T or 1 are used for true propositions
and F or 0 are assigned to false propositions.

5. BOARD
 a proposition that conveys one thought with no
connecting words
Simple Proposition
 contains two or more simple propositions that
are put together using connective words
Compound Proposition
Topic 1: Simple and Compound Proposition
Simple Proposition
Rowena is passing in Mathematics.
Compound Proposition
Rowena is passing in Mathematics but
she is failing in Social Science.

6. BOARD
 used to combine compound propositions using
words such as and, or, not, if … then, and if
and only if
Topic 2: Operations on Propositions
Conjunction
Disjunction
Conditional
Biconditional
Negation
 also known as propositional connectives
 process of joining connectives to form a
compound proposition

7. Topic 2: Operations on Propositions BOARD
 a statement that is false whenever the given statement is true, and true whenever
the given statement is false
Herbert is not good.
It is not the case that Herbert is good.
 obtained by inserting the word not in the given statement or by prefixing it with
phrases such as “It is not the case that…”
Herbert is good.
Some bottles have no labels.
Not all bottles have labels.
All bottles have labels.
Some students in uniform can enter the Internet cafe.
No student in uniform can enter the Internet cafe.
No participants are more active than the organizers.
Some participants are more active than the organizers.

8. Topic 2: Operations on Propositions BOARD
 two simple propositions connected using the word and
 sometimes the word but will be used in place of and in a given sentence
 two simple propositions connected using the word or
Today is Friday and tomorrow is Saturday.
Roel was on time, but Tom was late.
Roel was on time and Tom was late.
I will pass the Math exam or I will be promoted.

9. Topic 2: Operations on Propositions BOARD
 two simple propositions that are connected using the words if…then
antecedent (hypothesis) – the statement between the if and then
 conjunction of two conditional statements where the antecedent and consequent of
the first statement have been switched in the second statement
If you will recite the poem, then you will pass the oral examination.
If two sides of a triangle are congruent, then the angle opposite them are congruent,
and if two angles of a triangle are congruent, then the sides opposite them are congruent.
consequent (conclusion) – the sentence that follows then
Two sides of a triangle are congruent if and only if two angles opposite them are congruent.
 the abbreviation for if and only if is iff
If you will recite the poem, you will pass the oral examination.
a.
You will pass the oral examination if you will recite the poem.
b.

10. BOARD
Topic 3: Logical Symbols
The proposition 𝑞 ⟶ 𝑝
is called the converse of 𝑝 ⟶ 𝑞.
The proposition ∼ 𝑝 ⟶∼ 𝑞
is called the inverse of 𝑝 ⟶ 𝑞.
The proposition ∼ 𝑞 ⟶∼ 𝑝
is called the contrapositive of 𝑝 ⟶ 𝑞.

11. BOARD
 used to simplify work in logic
Topic 3: Logical Symbols
Connective Symbol Type of Statement
and ∧ conjunction
or ∨ disjunction
not ∼, ¬ negation
if…then ⟶ conditional
if and only if ⟷ biconditional
Sentential form
 a sequence of letters and/or
logical connectives that are
given such that when the
variables are replaced by
specific sentences, a
proposition is formed
Propositional Variables/
Sentential Variables
 lower case letters such as
p, q or r that are use to
denote a proposition
To define a proposition, say p, we usually write:
p : <the given statement>.
For instance, p : The earth has two moons.

12. Topic 3: Logical Symbols BOARD
Let p : He has green thumb.
q : He is a senior citizen.
Example 1:
Convert each compound proposition into symbols.
He has green thumb.
𝒑 ∧ 𝒒 He is a senior citizen.
and
He does not have green thumb.
∼ 𝒑 ∨∼ 𝒒 He is not a senior citizen.
or
4. If he has green thumb, then he is not a senior citizen.
1. He has green thumb and he is a senior citizen.
2. He does not have 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.
or
~(𝒑 ∨ 𝒒)
He is a senior citizen.
It is not the case He has green thumb
If he has green thumb,
𝒑 ⟶∼ 𝒒 He is not a senior citizen.
then

13. Topic 3: Logical Symbols BOARD
Let p : Robin can swim.
q : Tom plays the guitar.
Example 2:
Write each symbolic statement in words.
Robin can swim or Tom plays the guitar.
Robin can swim and Tom cannot play the guitar.
It is not the case that Robin can swim or Tom can play the guitar.
1. 𝒑 ∨ 𝒒
2. 𝒑 ∧∼ 𝒒
3. ∼ (𝒑 ∨ 𝒒)
4. ∼ (𝒑 ∧ 𝒒) It is not the case that Robin can swim and Tom can play the guitar.
Note that ~ 𝒑 ∨ 𝒒 means the negation of the entire statement 𝒑 ∨ 𝒒.
In ~𝒑 ∨ 𝒒, only statement 𝒑 is negated.
∼ (𝒑 ∨ 𝒒) means ∼ 𝒑 ∧∼ 𝒒
3. ∼ (𝒑 ∨ 𝒒) Robin cannot swim and Tom cannot play the guitar.
∼ (𝒑 ∧ 𝒒) means ∼ 𝒑 ∨∼ 𝒒
4. ∼ (𝒑 ∧ 𝒒) Robin cannot swim or Tom cannot play the guitar.

14. Topic 3: Logical Symbols BOARD
Example 3:
(𝒘 ∨ 𝒖) ∧ 𝒍
Note that the use of commas indicates which simple statements are grouped together.
Write the following in symbolic form:
1. Arnold is a working student (w) or under 25 years old (u), and lives in Manila (l).
2. Arnold is a working student (w), or under 25 years old (u) and lives in Manila (l).
𝒘 ∨ (𝒖 ∧ 𝒍)

15. Topic 3: Logical Symbols BOARD
Example 4:
p:Japeth Aguilar is a football player.
q:Japeth Aguilar is a basketball player.
r:Japeth Aguilar is a rock star.
s:Japeth Aguilar plays for the Ginebra.
• Japeth Aguilar is a football player or a basketball player, and he is not a rockstar.
• If Japeth Aguilar is a basketball player and a rockstar, then he is not a football
player
• Japeth Aguilar is a basketball player, if and only if he is not a football player
and he is not a rock star.
• It is not true that, Japeth Aguilar is a football player or a rock star.

16. (𝒒 ∧ 𝒓) ⟶ ¬𝒑
𝒓 ∧ ¬𝒒 ∨ 𝒑
¬(𝒑 ∨ 𝒓)
Topic 3: Logical Symbols BOARD
Example 4:
p:Japeth Aguilar is a football player.
q:Japeth Aguilar is a basketball player.
r:Japeth Aguilar is a rock star.
s:Japeth Aguilar plays for the Ginebra.

17. BOARD
 a table that shows the truth value of a
compound statement for all possible
truth values of its simple statements
 Truth value of a simple statement
is either true (T) or false (F)
 Truth value of a compound statement
depends on the truth values of its simple
statements and its connectives

18. BOARD
∼ 𝑝 is false when 𝑝 is true,
and
∼ 𝑝 is true when 𝑝 is false.
Topic 4: Negations
Truth Table for a
Simple Proposition’s
Negation
(∼ 𝒑)
𝒑 ∼ 𝒑
T
F
The negation of the negation of a statement is
the original statement.
Thus ∼ (∼ 𝑝) can be replaced by 𝑝
in any statement.

19. The conjunction 𝑝 ∧ 𝑞 is true
when 𝑝 and 𝑞 are both true;
otherwise, the conjunction is false
𝒑 𝒒
T T
T F
F T
F F
𝒑 ∧ 𝒒
Truth Table for the Conjunction p and q
𝒑 ∧ 𝒒
Truth Table for the Disjunction p or q
𝒑 ∨ 𝒒
Inclusive disjunction is false unless
both components are false.
𝒑 𝒒
T T
T F
F T
F F
𝒑 ∨ 𝒒

20. 𝑝⟶𝑞 is false only
when 𝑝 is true and 𝑞 is false; otherwise,
it is true
𝒑 𝒒
T T
T F
F T
F F
Truth Table for Conditional Statement
(𝒑⟶𝒒)
Truth Table for Biconditional Statement
(𝒑⟷𝒒)
𝑝⟷𝑞 is true when and only when 𝑝
and 𝑞 have the same truth value.
𝒑 𝒒
T T
T F
F T
F F
𝒑 ↔ 𝒒
𝒑 ⟶ 𝒒

21. BOARD
1. Recall dominant connectives and the use of
parentheses.
Truth Tables
2. Complete the columns under
 The simple statements (p,q, …)
 The connective negations inside parentheses
 Any remaining statements and their negations
 Any remaining connectives
reaching the final column.
3. The truth table of a compound proposition of n
component statements, each represented by a
different letter, has 2𝑛
number of rows.
1. TAUTOLOGY – ALL the
values in the FINAL
COLUMN are TRUE.
2. CONTRADICTION or
ABSURDITY- ALL the values
in the FINAL COLUMN are
FALSE.
3. CONTINGENCY- There is
at least ONE ROW FROM
THE LAST COLUMN where
it is true or false.

22. Truth Tables BOARD
Example 1:
𝒑 𝒒
T T
T F
F T
F F
∼ 𝒑 ∧ 𝒒 ∼ (∼ 𝒑 ∧ 𝒒)
𝒑

23. Truth Tables BOARD
Example 2:
𝒑 𝒒 𝒓
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
∼ 𝒓
F
T
F
T
F
T
F
T
𝒑 ∧ 𝒒 (𝑝 ∧ 𝑞) ∧ (∼ 𝑟 ∨ 𝑞)
∼ 𝒓 ∨ 𝒒

24. Truth Tables BOARD
Example 3:
𝒑 𝒒
T T
T F
F T
F F
∼ 𝒒 𝒑 ∧ 𝒒 𝒑 ⟶∼ 𝒒 (𝑝∧𝑞)⟷(𝑝⟶∼𝑞)

25. Truth Tables BOARD
𝒑 𝒒 𝒓
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
∼ 𝒑
F
F
F
F
T
T
T
T
(𝑞 ∧ 𝒓) →∼ 𝒑
𝒒 ∧ 𝒓
(𝒒 ∧ 𝒓) ⟶ ¬𝒑
Example 4:

26. Problem Set
BOARD
Problem Set 7 is already posted on our Google
Classroom.
Instructions:
1. Write your solutions and answers on clean
sheets of white short bond papers.
2. Use only pens with blue or black ink.
3. Scan your solutions sheet(s) in one (.pdf)-
format file and edit the filename as
(Surname.Firstname.pdf)
4. Submit your solutions by uploading the file
through Google Classroom before the
deadline.

27. References
BOARD
Oronce, Orlando. (2016). General mathematics (1st
Edition). Rex Book Store, Inc.
Lim, Yvette., et. al. (2016). Math for engaged
learning. Sibs Publishing House, Inc.
Rosal, A. Foundations of abstract mathematics.

28. Next Lesson
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MATHEMATICS OF A GRAPH