SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
National Highway, Crossing Rubber, Tupi, South Cotabato
COLLEGE OF TEACHERS EDUCATION
LEARNING MODULE
FOR
M104: LOGIC AND SET THEORY
WEEK 1
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 1 of 22
COURSE OUTLINE
COURSE CODE
:
M104
TITLE
:
LOGIC AND SET THEORY
TARGET POPULATION
:
1ST YEAR MATHEMATICS STUDENTS
INSTRUCTOR
:
MS. FARRAH M. LUCERO
Overview:
The course is a study of mathematical logic which covers topics such as propositions, logical operators,
rules of replacement, rules of inference, algebra of logic and quantifiers. It also includes a discussion
of elementary theory of sets such as fundamental concepts of sets, set theorems and set operations.
General Objective:




Explain and illustrate clearly, accurately, and comprehensively the basic mathematical
concepts, using relevant examples as needed
Show the connections between mathematical concepts that are related to one another
Demonstrate in detail basic mathematical procedures
Provide examples to illustrate the application of mathematical concepts and procedures.
The following are the topics to be discussed
Week 1: Logic statement and quantifiers
Week 2: Connectives and their truth value and truth table
Conditional statements and Variation of a conditional
Week 3: Tautologies and Contradiction
Week 4: Logical Equivalence and Laws of logic
Week 5: Logical Arguments and Venn diagram
Instruction to the Learners
Each chapter in this module contains a major lesson involving Logic and Set Theory. The units are
characterized by continuity, and are arranged in such a manner that the present unit is related to the
next unit. For this reason, you are advised to read this module. After each unit, there are exercises to
be given.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 2 of 22
WEEK 1
LOGIC
Logic is the language of reason. It is a set of rules we observe when we wish to reason out rationally.
In logic, we are interested to establish the truth or lack of truth of statements.
Mathematical Logic is the branch of mathematics which is mainly concerned with the relationship
between semantic concepts and syntactic concepts. A distinctive feature is its role in the foundations
of Mathematics, especially concerning whether the truth value of mathematical statements can be
obtained algorithmically within various axiom systems.
Symbolic logic studies some parts and relationships of the natural language by representing them
with symbols. The main ingredients of symbolic logic are statements and connectives.
Statements or propositions- the basic building block of logic. It is a declarative sentence that is either
True or False, but not both.
Examples:
1. It is sunny today.
2. Ms. W. will have a broader audience next month.
3. I did not join the club.
The three examples above are propositions because it can be identified as true or false.
Example:
1. How’s the weather?
2. Cool!
3. If I could…
The three examples above are not proposition because you cannot tell if it is true or false.
Simple Statement – a statement that conveys a single idea
Compound statement – a statement that conveys two or more ideas, and they are connected by words
and phrases such as and, or, if-then, and if-and-only-if
LOGICAL CONNECTIVES AND SYMBOLS
STATEMENT
CONNECTIVE
SYMBOLS
not p
p and q
p or q
If p, then q
p if and only if q
not
And
Or
If…then
If and only if
~p
p˄q
p˅q
p→q
p↔q
TYPE OF
STATEMENT
Negation
Conjunction
Disjunction
Conditional
Biconditional
Example:
1. Consider the following statements.
p: x is a rational number and q: x is a multiple of 2
a. Write in math language statements p and q.
𝑝: 𝑥 ∈ ℚ, 𝑞: 𝑥 = 2𝑛, 𝑛 ∈ ℤ
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 3 of 22
b. Write in math language the negation of p
𝑝: 𝑥 ∉ ℚ
c. Write in math language the compound statement p v q
𝑝 ˅ 𝑞: 𝑥 ∈ ℚ ˅ (𝑥 = 2𝑛, 𝑛 ∈ ℤ)
d. Write in math language the compound statement q → p
𝑞 → 𝑝: (𝑥 = 2𝑛, 𝑛 ∈ ℤ) → 𝑥 ∈ ℚ
Note: 𝑥 ∈ ℚ means x is a rational number
(𝑥 = 2𝑛, 𝑛 ∈ ℤ) is the definition of a number being a multiple of 2
𝑥 ∉ ℚ means x is not a rational number or x is an irrational number
2. INTERPRET MATH LANGUAGE
Let p, q and r be statements defined in the following manner.
𝑝: 𝑥 = 2𝑛 − 1, 𝑛 ∈ ℕ
(note: 2𝑛 − 1, 𝑛 ∈ ℕ define an odd number)
𝑞: 𝑥 ∈ 𝑃, 𝑤ℎ𝑒𝑟𝑒 𝑃 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
𝑟: 𝑥 > 2
Write the following statements in ordinary verbal sentence:
a. ~p
Number x is not a prime numbers.
b. (𝑞 ˄ 𝑟) → 𝑝
If x is a prime number greater than 2, then x is an odd integer.
c. 𝑝˄ 𝑞
Number x is a positive odd integer and it is a prime number.
d. (𝑝˄𝑞) → 𝑟
If x is a positive odd integer and prime, then it is greater than 2.
QUANTIFIERS
In Mathematics, quantifiers describe the quantity of objects in the domain of functions or simply the
quantity of objects in a set that satisfies some defined characteristics.
 UNIVERSAL QUANTIFIERS ( ∀ )
 For all
 For all, all and every assert that each member of a given
 All
set satisfies some condition
 Every
 None and no deny the existence of objects that meet a
 None
specified property
 No
 EXISTENTIAL QUANTIFIERS ( ∃ )
 Some
 It assert the existence of at least one object or element of a
 There exists
set that satisfies some specified property or condition.
 At least one
Examples:
1. Write Universal quantifiers into math language
a. For all integers x and y, x + y is an integer.
∀𝑥, 𝑦 ∈ ℤ
𝑥+𝑦 ∈ ℤ
∀𝑥∀𝑦 ∈ ℤ, (𝑥 + 𝑦) ∈ ℤ
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 4 of 22
b. For all real numbers x and y, if x+y=0, then x=-y
∀𝑥, 𝑦 ∈ ℝ
∀𝑥∀𝑦 ∈ ℝ, (𝑥 + 𝑦 = 0) → 𝑥 = −𝑦
c. Let x and y be real number not equal to zero. If xy=1, then y= 𝑥 −1
∀𝑥, 𝑦 ∈ ℝ
∀𝑥∀𝑦 ∈ ℝ, (𝑥 ≠ 0 ˄ 𝑦 ≠ 0), (𝑥𝑦 = −1 → 𝑦 = 𝑥 −1
2. Write existential quantifiers into math language
a. For all non-zero real number x, there is a real number y such that xy=1.
∀𝑥 ∈ ℝ, (𝑥 ≠ 0)
∃𝑦 ∈ ℝ
∀𝑥 ∈ ℝ, (𝑥 ≠ 0), ∃𝑦 ∈ ℝ (𝑥𝑦 = 1)
b. For any real number x and y, if x>y, then there is a real number w such that x>w>y
∀𝑥, 𝑦 ∈ ℝ
∃𝑤 ∈ ℝ
∀𝑥∀𝑦 ∈ ℝ, [(𝑥 > 𝑦) → ∃𝑤 ∈ ℝ, (𝑥 > 𝑤 > 𝑦)]
c. Let a, b and c be real numbers such that 𝑎 ≠ 0. There exists a real number x that satisfies
𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0 if and only if 𝑏2 − 4𝑎𝑐 ≥ 0.
∀𝑎∀𝑏∀𝑐 ∈ ℝ, (𝑎 ≠ 0), ∃𝑥 ∈ ℝ [(𝑎𝑥 2 + 𝑏𝑥 + 𝑐 = 0) → (𝑏2 − 4𝑎𝑐 ≥ 0)]
NEGATION OF QUANTIFIERS
If a statement is true for any integer, its negation means there is at least one integer where the
statement does not apply. Further, if there is an object that satisfies a specified condition, its negation
means there are no object that meet that condition. We use ~ (tilde) to indicate negation.
STATEMENT
All X are Y
No X are Y
Some X are not Y
Some X are Y
NEGATION
Some X are not Y
Some X are Y
All X are Y
No X are Y
Let P(x) be the statement about a variable x which could be either true or false.
STATEMENT
NEGATION
RESULT
(
)
∀𝑥 𝑃(𝑥)
~[∀𝑥 𝑃 𝑥 ]
∃𝑥 ~𝑃(𝑥)
∀𝑥 ~𝑃(𝑥)
~[∀𝑥 𝑃 (𝑥 )]
∃𝑥 𝑃(𝑥)
∃𝑥~ 𝑃(𝑥)
~[∃𝑥~ 𝑃(𝑥 )]
∀𝑥 𝑃(𝑥)
∃𝑥 𝑃(𝑥)
~[∃𝑥𝑃(𝑥 )]
∀𝑥~ 𝑃(𝑥)
Example:
1. Write in mathematical language the negation of each of the statements and translate them into
English sentence
a. ∀𝑥∀𝑦 ∈ ℤ, (𝑥 + 𝑦) ∈ ℤ
 ~[∀𝑥∀𝑦 ∈ ℤ, (𝑥 + 𝑦) ∈ ℤ] ≡ ∃𝑥∃𝑦 ∈ ℤ, (𝑥 + 𝑦) ∉ ℤ
There are integers x and y such that their sum is not an integer.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 5 of 22
b. ∀𝑥∀𝑦 ∈ ℝ(𝑥 ≠ 0 ˄ 𝑦 ≠ 0), (𝑥𝑦 = 1 → 𝑦 = 𝑥 −1 )
 ~[∀𝑥∀𝑦 ∈ ℝ(𝑥 ≠ 0 ˄ 𝑦 ≠ 0), (𝑥𝑦 = 1 → 𝑦 = 𝑥 −1 )] ≡
∃𝑥∃𝑦 ∈ ℝ(𝑥 ≠ 0 ˄ 𝑦 ≠ 0), (𝑥𝑦 = 1 → 𝑦 ≠ 𝑥 −1 )
There are integers x and y both not equal to zero such that 𝑥𝑦 = 1 but 𝑦 ≠ 𝑥 −1 .
c. ∀𝑥 ∈ ℝ (𝑥 ≠ 0), ∃𝑦 ∈ ℝ(𝑥𝑦 = 1)
 ~[∀𝑥 ∈ ℝ (𝑥 ≠ 0), ∃𝑦 ∈ ℝ(𝑥𝑦 = 1) ≡ ∃𝑥 ∈ ℝ(𝑥 ≠ 0), ~∃𝑦 ∈ ℝ (𝑥𝑦 = 1)
There is a real number x not equal to zero for which no real number y exists such that xy=1.
ACTIVITY 1
Directions: Read and understand this module. Provide what is asked. Write your answer in a whole
sheet of bond paper.
1. Write in mathematical language the negation of each of the statements and translate them into
English sentence. (10 points each)
a. ∀𝑥 ∈ ℤ, ∃𝑦 ∈ ℤ(𝑥 + 𝑦 = 0)
b. ∀𝑥 ∈ ℝ (𝑥 2 + 1 = 0)
c. ∀𝑥 ∈ ℕ, ∀𝑦 ∈ ℕ (𝑥 + 𝑦 = 0)
d. ∀𝑥 ∈ ℤ, (∀𝑦 ∈ ℤ(𝑦𝑥 = 1)
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 6 of 22
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
National Highway, Crossing Rubber, Tupi, South Cotabato
COLLEGE OF TEACHERS EDUCATION
___________________________________________________
LEARNING MODULE
FOR
M104: LOGIC AND SET THEORY
_____________________________________________________
WEEK 2
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 7 of 22
WEEK 2
CONNECTIVES AND TRUTH VALUES
Connectives - join simple statements into more complex statements, called compound statements.
Truth values
-
Every Logical statement, simple or compound, is either true or false. When a statement is true
it is denoted by T and if a statement is false it is denoted by F.
Truth tables
-
-
-
-
A truth table is a tabulation of possible truth values of a statement depending on the value of its
constituent statements. We shall resort to truth tables often when finding the truth value of a
compound statements.
For a given statement P, we will look at the possible truth values of p. Namely, either p is true or
p is False.
P
T
F
If we are analyzing two statements p and q at the same time, we need to consider all possible
combinations of truth values
P
q
T
T
T
F
F
T
F
F
If we are analyzing three statements p, q and r, there are 8 possible combinations since the
three statements can be either be TRUE or FALSE.
P
q
r
T
T
T
T
F
T
F
T
T
F
F
T
T
T
F
T
F
F
F
T
F
F
F
F
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 8 of 22
CONNECTIVES
 Negation
- The negation (NOT) of a statement p is indicated by ~p. We read it as “not p”. The truth value
for the negation of a statement is as follows.
- If p is true, then ~p is false. If p is false, then ~p is true.
p
~p
T
F
F
T
 Conjunction
- Given two statements, p and q, the connective AND, also called conjunction, is denoted
by 𝑝 ^ 𝑞.
-
𝑝 ^ 𝑞 is true only when both p and q are true.
p
q
𝒑^𝒒
T
T
T
T
F
F
F
T
F
F
F
F
 Disjunction
- Given two statements p and q the connective OR, also called disjunction, is denoted by 𝑝 ˅ 𝑞
- 𝑝 ˅ 𝑞 is true in all cases except when both p and q are false. This is equivalent to saying that
𝑝 ˅ 𝑞 is true if at least one of the statement is true.
p
q
𝒑˅𝒒
T
T
T
T
F
T
F
T
T
F
F
F
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 9 of 22
 Implication (if- then)
- Also called conditional Statements
- Given two statements p and q the connective IMPLIES or IMPLICATION is denoted by 𝑝 → 𝑞.
This is read as “p implies q” or “if p then q”.
- In Implication, statement p is called hypothesis or antecedent or premise. The statement q is
called the conclusion or consequence.
- 𝑝 → 𝑞 is true in all cases except if p is true and q is false.
-
It is False only when a true hypothesis leads to a false conclusion.
P
q
𝒑→𝒒
T
T
T
T
F
F
F
T
T
F
F
T
 Biconditional Statement (if-and only-if)
- The connective IF AND ONLY IF (iff) is a conjunction of two implications. This is denoted by 𝑝 ↔
𝑞. As a conjunction of two implications, it is equivalent to the compound statement:
(𝑝 → 𝑞)˄(𝑞 → 𝑝).
- 𝑝 ↔ 𝑞 is true when both p and q is true or when both p and q are false.
P
q
𝒑↔𝒒
T
T
T
T
F
F
F
T
F
F
F
T
VARIATIONS OF CONDITIONAL STATEMENT
For two given statements p and q, various conditional statements can be formed. The following
are the three cases:
a. CONVERSE: the converse of a conditional statement “if p then q” is “if q then p”. symbolically
converse of 𝒑 → 𝒒 𝒊𝒔 𝒒 → 𝒑
b. INVERSE: the inverse of the conditional “if p then q” is “if not p then not q”. Symbolically inverse
of 𝒑 → 𝒒 𝒊𝒔 ~𝒑 → ~𝒒
c. CONTRAPOSITIVE: the contrapositive of a conditional “if p then q” is “if not q then not p”.
Symbolically contrapositive of 𝒑 → 𝒒 𝒊𝒔~𝒒 → ~𝒑
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 10 of 22
Example:
1. All numbers that are multiples of four are multiples of eight.
Conditional:
If a number is a multiple of 4, then it is a multiple of 8.
Converse:
If a number is a multiple of 8, then it is a multiple of 4.
Inverse:
If a number is not a multiple of 4, then it is not a multiple 8.
Contrapositive:
If a number is not a multiple of 8, then it is not a multiple of 4.
2. All painters are creative.
Conditional:
If you are a painter, then you are creative.
Converse:
If you are creative, then you are a painter.
Inverse:
If you are not a painter, then you are not creative.
Contrapositive:
If you are not creative, then you are not a painter.
ACTIVITY 2
Directions: Read and understand this module. Provide what is asked. Write your answer in a whole
sheet of bond paper.
1. Give the conditional, converse, inverse and contrapositive in the if-then form of the following
statements. (10 points each)
a. All even numbers are divisible by two.
b. All Filipinos are hospitable.
c. Regular polygons are equiangular.
d. Isosceles triangles have two equal sides.
e. Squares have four equal sides.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 11 of 22
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
National Highway, Crossing Rubber, Tupi, South Cotabato
COLLEGE OF TEACHERS EDUCATION
LEARNING MODULE
FOR
M104: LOGIC AND SET THEORY
WEEK 3
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 12 of 22
WEEK 3
TAUTOLOGY AND CONTRADICTION
Tautology is a statement which is always true.
Contradiction is a statement that is always false.
Example:
1. Either Lolet is present or absent in her class
Solution: let p: Lolet is present
P
~p
𝒑˅~𝒑
T
F
T
F
T
T
Step 1: find the negation of p
Step 2: find the truth values of 𝒑˅~𝒑
Thus, this statement is a tautology.
2. 𝒑 → (𝒒 → 𝒑)
P
q
q→p
T
T
T
T
F
T
F
T
F
F
F
T
Step 1: find the truth values of q → p
Step 2: find the truth values of 𝑝 → (𝑞 → 𝑝)
Thus, this statement is a tautology.
3. 𝒑 ˄(~𝒑˄𝒒)
P
q
~p
T
T
F
T
F
F
F
T
T
F
F
T
Step 1: find the negation of p
Step 2: find the truth values of ~p ˄ q
Step 3: find the truth values of 𝑝 ˄(~𝑝˄𝑞)
Thus, this statement is a contradiction.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
𝒑 → (𝒒 → 𝒑)
T
T
T
T
~p ˄ q
F
F
T
F
𝒑 ˄(~𝒑˄𝒒)
F
F
F
F
Page 13 of 22
ACTIVITY 3
Directions: Read and understand this module. Provide what is asked. Write your answer in a whole
sheet of bond paper.
1. Make a truth table for the following statement then determine if it is tautology, contradiction or
neither. (10 points each)
a. ~(𝑝˅~𝑝)
b. (𝑝 → 𝑞 ) → (𝑞 → 𝑝)
c. 𝑝 ˅ (~(𝑝˄~𝑞))
d. 𝑝˄(~𝑝˄𝑞)
e. ~[𝑝˅(~𝑝˅𝑞)]
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 14 of 22
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
National Highway, Crossing Rubber, Tupi, South Cotabato
COLLEGE OF TEACHERS EDUCATION
LEARNING MODULE
FOR
M104: LOGIC AND SET THEORY
WEEK 4
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 15 of 22
WEEK 4
LOGICAL EQUIVALENCE
LOGICAL EQUIVALENCE
-
Means that two sides always have the same truth values.
Symbol: ⇔ 𝑜𝑟 ≡ (we’ll use ⇔)
Example:
1. Prove that 𝑝 → 𝑞 ⇔ ~𝑝 ˅ 𝑞 are logically equivalent.
𝑝 → 𝑞 ⇔ ~𝑝 ˅ 𝑞
P
T
T
F
F
Q
T
F
T
F
𝑝→𝑞
T
F
T
T
~𝑝
F
F
T
T
~𝑝 ˅ 𝑞
T
F
T
T
Step 1: Find the truth values of LHS (left-hand side), 𝑝 → 𝑞
Step 2: find the negation of p
Step 3: find the truth values of the RHS (right-hand-side), ~𝑝 ˅ 𝑞
Step 4: if the truth values of the LHS and the RHS are the same then they are logically equivalent.
Since the truth values of 𝒑 → 𝒒 (LHS) and ~𝒑 ˅ 𝒒 (RHS) are the same, then they are
logically equivalent.
2. 𝑝 → 𝑞 ⇔ ~𝑞 → ~𝑝
P
T
T
F
F
Q
T
F
T
F
𝑝→𝑞
T
F
T
T
~𝑝
F
F
T
T
~𝑞
F
T
F
T
~𝑞 → ~𝑝
T
F
T
T
Step 1: Find the truth values of LHS (left-hand side), 𝑝 → 𝑞
Step 2: find the negation of p and negation of q
Step 3: find the truth values of the RHS (right-hand-side), ~𝑞 → ~𝑝
Step 4: if the truth values of the LHS and the RHS are the same then they are logically equivalent.
Since the truth values of 𝒑 → 𝒒 (LHS) and ~𝒒 → ~𝒑(RHS) are the same, then they are
logically equivalent.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 16 of 22
LAWS OF LOGIC
The table shows known logically equivalent statement, it can be verified or proved with a truth
table. We can use the following laws of logic to prove tautologies, contradiction and logical
equivalence.
Tautology
Contradiction
Idempotence
Commutative
Associative
Distributive
Double Negation
DeMorgan’s Laws
Absorption
Dominance
1 ( statement is always true)
0 ( statement is always false)
𝑝˅𝑝 ⇔ 𝑝
𝑝˄𝑝 ⇔ 𝑝
𝑝˅𝑞 ⇔ 𝑞˅𝑝
𝑝˄𝑞 ⇔ 𝑞˄𝑝
(𝑝˄𝑞)˄𝑟 ⇔ 𝑝˄(𝑞˄𝑟)
(𝑝˅𝑞)˅𝑟 ⇔ 𝑝˅(𝑞˅𝑟)
𝑝 ˅ (𝑞˄𝑟) ⇔ (𝑝˅𝑞 )˄(𝑝˅𝑟)
𝑝 ˄ (𝑞˅𝑟) ⇔ (𝑝˄𝑞 )˅(𝑝˄𝑟)
~(~𝑝) ⇔ 𝑝
~(𝑝˅𝑞) ⇔ ~𝑝˄~𝑞
~(𝑝˄𝑞) ⇔ ~𝑝˅~𝑞
𝑝˄1 ⇔ 𝑝
𝑝˄0 ⇔ 0
𝑝˅1 ⇔ 1
𝑝˅0 ⇔ 𝑝
Examples:
1. Simplify: ~(~𝑝 → 𝑞)˅(𝑝˄~𝑞)
⇔~(~~𝑝˅𝑞)˅(𝑝˄~𝑞)
⇔~(𝑝˅𝑞)˅(𝑝˄~𝑞)
⇔(~𝑝˄~𝑞)˅(𝑝˄~𝑞)
⇔ ( 𝑝˄~𝑝)˅~𝑞
⇔ 0˅~𝑞
⇔ ~𝑞
known tautology
double Negation
DeMorgan
Distributive
Known Contradiction
Absorbtion
2. Prove that ~𝑞˅(𝑝 → 𝑞) is a tautology.
Proof:
~𝑞˅(𝑝 → 𝑞) ⇔ ~𝑞˅(~𝑝˅𝑞)
Known L.E
⇔~𝑞˅(𝑞˅~𝑝)
Commutative
⇔(~𝑞˅𝑞)˅~𝑝
Associative
⇔1˅~𝑝
known tautology
⇔1
Dominance
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 17 of 22
3. Show that (𝑝˄𝑞)˄[(𝑞˄~𝑟)˅(𝑝˄𝑟)] ⇔ ~(𝑝 → ~𝑞)
Proof:
(𝑝˄𝑞)˄[(𝑞˄~𝑟)˅(𝑝˄𝑟)]
⇔[(𝑝˄𝑞)˄(𝑞˄~𝑟)]˅[(𝑝˄𝑞)˄(𝑝˄𝑟)]
Distributive
⇔[((𝑝˄𝑞)˄𝑞)˄~𝑟)]˅[((𝑝˄𝑞 )˄𝑝)˄𝑟]
Associative
⇔[(𝑝˄(𝑞˄𝑞 ))˄~𝑟]˅[((𝑝˄𝑝)˄𝑞)˄𝑟]
Commutative, Associative
⇔[(𝑝˄𝑞)˄~𝑟]˅[(𝑝˄𝑞)˄𝑟]
Idempotence
⇔(𝑝˄𝑞)˄(~𝑟˅𝑟)
Distributive
⇔(𝑝˄𝑞)˄1
Known Tautology
⇔(𝑝˄𝑞)
Absorbtion
⇔~~(𝑝˄𝑞)
Double Negation
⇔~(~𝑝˅~𝑞)
Demorgan
⇔~(𝑝 → 𝑞)
Known L.E
ACTIVITY 4
Directions: Read and understand this module. Provide what is asked. Write your answer in a whole
sheet of bond paper.
1. Use the truth table method to verify whether the following logical consequences and
equivalences are correct. (10 points each)
a. 𝑝 ↔ 𝑞 ⇔ (𝑝 → 𝑞)˄(𝑞 → 𝑝)
b. ~(𝑝˅𝑞) ⇔ ~𝑝 → ~𝑞
2. Deteremine whether the following statements are tautology or contradiction using the laws of
logic: (10 points each)
a. 𝑝 → (𝑞 → 𝑝)
b. 𝑝 ˄(~𝑝˄𝑞)
c. 𝑝 ˅ (~(𝑝˄~𝑞))
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 18 of 22
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
National Highway, Crossing Rubber, Tupi, South Cotabato
COLLEGE OF TEACHERS EDUCATION
LEARNING MODULE
FOR
M104: LOGIC AND SET THEORY
WEEK 5
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 19 of 22
WEEK 5
LOGICAL ARGUMENTS AND VENN DIAGRAMS
A logical arguments is a list of premises (such as assumptions, rules, facts and observations),
followed by a single statement called the conclusion. Here are two examples of logical
arguments:
All giraffes have long necks
I have a pet giraffe
Therefore, one of my pets has a long neck
Most kids love chocolate
I am no longer a kid
Therefore, I hate chocolate
Notice that each of these two arguments has two premises (the first two statements) and one
conclusion, preceded by the word therefore.
Validity of a logical argument
A logical argument is valid if, whenever all premises are true, then the conclusion is also true.
A logical argument containing quantifiers is called a categorical syllogism. Recall that quantifiers
are terms such as all, every, each, none (the universal quantifiers), and some, most, at least
one, there is (the existential quantifiers). The validity of a categorical syllogism can be tested in
a particularly simple way using Venn diagrams.
EXAMPLE 1:
All giraffes have long necks
I have a pet giraffe
Therefore, one of my pets has a long neck
P1: All giraffes have long necks.
P2: I have a pet giraffe.
STEP 1: “draw” the first premise using two Venn diagrams: the first circle represents the group
of all giraffes, and the second circle represents the group of animals with long necks. Premise 1
tells us that the set of all giraffes is contained within the set of animals with long necks.
STEP 2: We complete the drawing by “adding” the second premise. This can be done simply by
placing a dot (representing my pet giraffe) inside the Venn diagram describing the group of all
giraffes (inner circle).
STEP 3: We check whether the drawing forces the conclusion to be true: obviously, since your
pet giraffe is inside the set of all giraffes, which is inside the set of animals with long necks, your
pet giraffe has to have a long neck too! So the conclusion (C: one of my pets has a long neck)
follows from the premises.
Step 4: We conclude that the first argument is valid.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 20 of 22
EXAMPLE 2:
Most kids love chocolate
I am no longer a kid
Therefore, I hate chocolate
P1: Most kids love chocolate.
P2: I am no longer a kid.
Premise 1 ‘Most kids love chocolate’ can be rephrased as ‘Some kids love chocolate’ without
substantially changing its meaning, Premise 2 can also be rephrased as ‘I am not a kid’. If we
represent Premise 1 using Venn diagrams, we obtain the two circles at the top. When drawing
the second premise, we notice that there are two possible options. The dot representing “I” must
be placed outside of the circle representing Kids, however it may be outside of (left bottom
diagram) or inside (right bottom diagram) the circle representing Chocolate lovers. In the first
case (outside), the conclusion is true, while for the second scenario the conclusion is clearly
false. Since we found a case in which true premises do not force a true conclusion, we conclude
that this argument is invalid.
EXAMPLE 3:
To show that an argument is invalid it is sufficient to draw one scenario representing each of the
premises, in which the conclusion is false. You will need to build up your intuition by looking at
several examples. A good one is:
Some marigolds are yellow.
All lemons are yellow.
Therefore, some lemons are marigolds.
Therefore, it is an invalid argument.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 21 of 22
ACTIVITY 5
Directions: Read and understand this module. Provide what is asked. Write your answer in a whole
sheet of bond paper.
1. Use Venn diagrams to determine whether the following logical arguments containing
quantifiers are valid or not valid. Make sure to first identify the premises and the conclusion.
(10 points each)
a. Every men is mortal.
Socrates is a man.
Therefore, Socrates is mortal.
b. Some philosophers are absent-minded.
Amanda is absent-minded.
Therefore, Amanda is a philosopher
c. All A’s are B’s.
Some B’s are C’s.
Therefore, some A’s are C’s.
d. All vitamins are healthy.
Caffeine is a vitamin.
Therefore, caffeine is healthy
e. No fish is a mammal.
Cows are mammals.
Therefore, cows are not fish.
M104: Logic and Set Theory
SOUTH EAST ASIAN INSTITUTE OF TECHNOLOGY, INC.
Page 22 of 22