Logic.pptx

ANSWER THE FOLLOWING QUESTIONS
1. Divide 30 by half and add 10. What do you get?
2. Imagine you are in a sinking rowboat surrounded by sharks. How would
you survive?
3. Eskimos are very good hunters, but they never hunt penguins. Why
not?
4. There was an airplane crash. Every single person died, but two people
survived. How is this possible?
5. If John’s son is my son’s father, what am I to John?

ANSWER THE FOLLOWING QUESTIONS
6. A clerk in a butcher shop stands five feet and ten inches tall. He wears size
10 shoes. What does he weigh?
7. In British Columbia, you cannot take a picture of a man with a wooden leg.
Why not?
8. 0, 3, 6, 9, 12, 15, 18, … ,960. How many numbers are in the series if all terms
are included?
9. Tina’s granddaughter is my daughter’s daughter. Who is Tina?
10. I recently returned from a trip. Today is Tuesday. I returned three days before
the day after the day before tomorrow. On what day did I return?

WHAT IS LOGIC?
 The branch of philosophy concerned with analyzing the patterns of reasoning
by which a conclusion is drawn from a set of premises, without reference to
meaning of context. – Collins English Dictionary

FOUNDATIONS OF LOGIC
Mathematical Logic is a tool for working with complicated compound
statements. It includes:
 A language for expressing them
 A concise notation for writing them.
 A methodology for objectively reasoning about their truth or falsity.
It is the foundation for expressing formal proofs in all branches of Mathematics.

PROPOSITIONAL LOGIC
 Propositional Logic is the logic of compound statements built from simpler
statements using so-called Boolean connectives.

DEFINITION OF PROPOSITION
 A proposition is simply a statement (i.e., a declarative sentence) with a
definite meaning, having a truth value that’s either true (T) of false (F) (never
both, neither, or somewhere in between).
 A proposition (statement) may be denoted by a variable like p, q, r, …, called
a proposition (statement) variable.

EXAMPLES OF PROPOSITIONS
1. Bangkok is the capital of Thailand.
2. A is a consonant.
3. William Shakespeare wrote Romeo and Juliet
4. The Catholic Bible has 73 books.
5. 6 + 9 = 15
6. 𝜋 is an irrational number.
7. Pneumonoultramicroscopicsilicovolcanoconiosis is the longest word in a
major dictionary.

PROPOSITION OR NOT?
1. Clean the room.
2. Scientifically, eggplant is not a vegetable.
3. Can’t leopard remove its own spots?
4. What a magnificent building!
5. Litotes is an exaggerated statement.
6. The Passion of Christ is directed by Mel Gibson.
7. x = 3
8. x + y > 4
9. Verb is a word that denotes an action.
10. This statement is false.

DETERMINE WHETHER FOLLOWING IS A PROPOSITION. IF IT IS A
PROPOSITION, STATE THE TRUTH VALUE.
1. The moon is made up of green
cheese.
2. August has 30 days.
3. 7 ∈ {1, 3, 5, 7, 9}
4. Leave me alone!
5. Why are you so upset?
6. 5 + 1 = 2
7. China is in South-east Asia.
8. 𝜋 > 3
9. n is a prime number.
10. The sum of two prime numbers is
even.
11. Write 5 propositions and their truth values. (5
points)

SIMPLE AND COMPOUND PROPOSITIONS
A simple statement or proposition is a statement containing no
connectives. In other words, a proposition is considered simple if it
cannot be broken up into sub-propositions.
Examples:
1. Today is Tuesday.
2. 8 + 9 > 25.

On the other hand, a compound proposition is made up of two or
more propositions joined by the connectives/operators.
Example: 2 is an even number and 2 is divisible by 4.
The lowercase English letters such as p, q, r, and s are used to
represent compound propositions.

If p is a proposition, then the truth value of p can be in any of the
two: True (T) of False (F), which is shown in the table below:
p
T
F

TRUTH TABLE
A truth table shows how the truth or falsity of a compound
statement depends on the truth or falsity of the simple statements
from which it's constructed.
It consists of columns for one or more input values, says, p and q and
one assigned column for the output results. The output which we get
here is the result of the unary or binary operation performed on the
given input values.

TRUTH TABLE
If there are two statements, p and q, then there are 4 possible truth
values in which T and F of p are matched to cases T and F of q.
p q
T T
T F
F T
F F

OPERATORS / CONNECTIVES
 An operator or connective combines one or more propositions
into a larger expression. (E.g., “+” in numeric expressions)
 Unary operators take 1 operand (e.g., −3)
 Binary operators take 2 operands (e.g., 3 × 4).
 Propositional or Boolean operators operate on propositions or
truth values instead of on numbers.

SOME POPULAR CONNECTIVES
Formal Name Nickname Arity Symbol
Negation operator NOT Unary ¬
Conjunction operator AND Binary ∧
Disjunction operator OR Binary ∨
Exclusive-OR operator XOR Binary ⊕
Implication operator IMPLIES Binary →
Biconditional operator IFF Binary

NEGATION
The Negation of a statement is the opposite of the given
mathematical statement. If “p” is a statement, then the negation of
statement p is represented by ~p. The symbols used to represent the
negation of a statement are “~” or “¬”.

NEGATION
For example, the given sentence is “Arjun’s dog has a black tail”.
Then, the negation of the given statement is “Arjun’s dog does not
have a black tail”. Thus, if the given statement is true, then the
negation of the given statement is false.
It is observed that the “negation of the negated sentence is the
original sentence”

NEGATION
Assume that the given sentence, p is “Triangle ABC is an equilateral
triangle”.
Thus, the negation of the given sentence, ~ p is “Triangle ABC is not
an equilateral triangle”.
The negation of the negated sentence ~(~ p) is “Triangle ABC is an
equilateral triangle”.
Hence, this proves that the negation of the negated sentence is the
given original sentence.

NEGATION
Determining the truth value of a negation proposition is easy
because the truth value of a negation is always opposite to the truth
value of the original proposition.
𝑝 ¬𝑝
T F
F T

NEGATION
Give the negation for the following sentences:
1. Line a is parallel to line b.
2. Some prime numbers are odd.
3. 3+3 = 6.
4. x is a real number such that x = 4.
5. 3 is a not a prime number.

NEGATION
Give the negation for the following sentences:
6. This book is not interesting.
7. Hua Hin is the capital of Thailand.
8. Food is not expensive in the United States.
9. 3 + 5 = 7.
10. The summer in South Korea is hot and sunny.

CONJUNCTION
A CONJUNCTION is a statement formed by adding two statements
with the connector AND. The symbol for conjunction is ‘∧’ which can
be read as ‘and’.
When two statements p and q are joined in a statement, the
conjunction will be expressed symbolically as p∧ q. If both the
combining statements are true, then this statement will be true;
otherwise, it is false.

RULES FOR A CONJUNCTION
 The conjunction statement will only be true if both the combining
statements are true otherwise, false.
 Let p and q be the two statements. The compound statement p ∧ q
is called the conjunction of p and q.
 The symbol “∧” that denotes the conjunction, it is read as “and”
which is the logical connective.

CONJUNCTION TRUTH TABLE
Let us make a truth table for P and Q, i.e. P ∧ Q.
P Q P ∧ Q
T T T
T F F
F T F
F F F
In this table, we can say that the conjunction is true only when both P
and Q are true. If they are not, then the conjunction statement will be
false.

CONJUNCTION EXAMPLES
1. Let r: 5 be a rational number and s: 15 be a prime number. Is it a
conjunction?
Solution:
Given that r: 5 is a rational number. This proposition is true.
s: 15 is a prime number. This proposition is false as 15 is a composite
number.
Therefore, as per the truth table, r and s is a false statement.
So, r ∧ s = F

2. Let a: x be greater than 9 and b: x be a prime number. Is it a conjunction?
Solution:
Since x is a variable whose value we don’t know. Let us define a range for a and
b.
To find the range let us take certain values for x;
When x= 6: a and b is false. Hence, a ∧ b is false.
When x= 3: a is false but b is true. But still, a ∧ b is false.
When x= 10: a is true but b is false. But still, a ∧ b is false.
When x= 11: a is true and b is true. Hence, a ∧ b is true.
Hence the conjunction a and b is only true when x is a prime number greater
than 9.

3. Let: 4 be a rational number and let 7 be a prime number. Is this a
Conjunction?
Solution:
Let statement p be that 4 is a rational number. Statement p is TRUE.
Let statement q be that 7 is a prime number. Statement q is TRUE
As per the Truth Table, if p is True and if q is also true, then “p ^ q” is
True
So, in our case, the Conjunction “p ^ q” that is “4 is a rational number,
and 7 is a prime number” is True.

4. A: The sun rises in the east B: It will definitely rain day after tomorrow. Is
this a true Conjunction?
Solution:
Statement A which states that the sun rises in the east is a True fact and
hence can never be changed. So, statement A is True.
Statement B has the possibility to be false or True. A prediction can never be
made with 100% surety that it will definitely rain the day after tomorrow.
Thus, statement B has both possibilities. But, for sure, it cannot be proved as
a totally True statement at present. Hence, statement B is False.
So, according to the Truth Table, the Conjunction A^B is False.

TRY THESE!
1. Given:
p: The number 11 is prime. true
q: The number 17 is composite. false
r: The number 23 is prime. true
Problem:
For each conjunction below, write a sentence and indicate if
it is true or false.
1. 𝑝 ∧ 𝑞
2. 𝑝 ∧ 𝑟
3. q∧ 𝑟

TRY THESE!
2. Construct a truth table for each conjunction below:
a. 𝑥 𝑎𝑛𝑑 𝑦
b. ¬𝑥 𝑎𝑛𝑑 𝑦
c. ¬𝑦 𝑎𝑛𝑑 𝑥

TRY THESE!
3. Given:
r: The number x is odd.
s: The number x is prime.
Problem:
Can we list all truth values for r ∧ s in a truth table? Why or why
not?

DISJUNCTION
 A disjunction is a compound statement formed by joining two
statements with the connector OR.
 The disjunction "p or q" is symbolized by “p ∨ q”.
 A disjunction is false if and only if both statements are false;
otherwise it is true.

DISJUNCTION
The truth values of p ∨ q are listed in the truth table below.
p q p ∨ q
T T T
T F T
F T T
F F F

EXAMPLES
Given:
p: Ann is on the softball team.
q: Paul is on the football team.
Problem: What does p v q represent?
Solution: Statement p represents, "Ann is on the softball team" and
statement q represents, "Paul is on the football team." The symbol v is
a logical connector which means "or." Thus, the compound statement p
v q represents the sentence, "Ann is on the softball team or Paul is on
the football team." The statement p v q is a disjunction.

EXAMPLES
Given:
a: A square is a quadrilateral.
b: Harrison Ford is an American actor.
Problem: Construct a truth table for the disjunction "a or b."

TRY THESE!
Given:
r: x is divisible by 2.
s: x is divisible by 3.
Problem: What are the truth values of r v s?

Given:
p: 12 is prime.
q: 17 is prime.
r: 19 is composite.
Problem:
Write a sentence for each disjunction below. Then
indicate if it is true or false.
1. p v q
2. p v r
3. q v r
TRY THESE!

Complete a truth table for each disjunction below.
1. a or b
2. a or not b
3. not a or b
TRY THESE!

TRY THESE!
Given:
x: Jayne played tennis.
y: Chris played softball.
Problem: Construct a truth table for conjunction “x and y” and
disjunction “x or y”.

CONDITIONAL STATEMENT/IMPLICATION
A conditional statement/implication is the compound statement
of the form “if p, then q .” It is denoted p→q , which is read as “p
implies q.”
It is false only when p is true and q is false and is true in all other
situations.

A truth table for p→q is shown below.
p q p→q
T
T
F
F
T
F
T
F
T
F
T
T

PARTS OF A CONDITIONAL STATEMENT
Hypothesis (if) and Conclusion (then) are the two main parts that
form a conditional statement.
Conditional Statement: If today is Monday, then yesterday was
Sunday.
Hypothesis: "If today is Monday."
Conclusion: "Then yesterday was Sunday."

Conditional statement examples:
Example 1: If a number is divisible by 4, then it is divisible by 2.
Example 2: If today is Monday, then yesterday was Sunday.

EXAMPLES
Given:
p: I do my homework.
q: I get my allowance.
Problem: What does p→q represent?
Solution: The sentence, "I do my homework" is the hypothesis and
the sentence, "I get my allowance" is the conclusion. Thus, the
conditional p→q represents the hypothetical proposition, "If I do my
homework, then I get an allowance." However, doing your homework
does not guarantee that you will get an allowance! In other words,
there is not always a cause-and-effect relationship between the
hypothesis and conclusion of a conditional statement.

EXAMPLES
Given:
a: The sun is made of gas.
b: 3 is a prime number.
Problem: Write a→b as a sentence. Then construct a truth
table for this conditional.

In the previous example, "The sun is made of gas" is the
hypothesis and "3 is a prime number" is the conclusion. Note that the
logical meaning of this conditional statement is not the same as its
intuitive meaning. In logic, the conditional is defined to be true unless
a true hypothesis leads to a false conclusion. The implication of a→b
is that: since the sun is made of gas, this makes 3 a prime number.
However, intuitively, we know that this is false because the sun and
the number three have nothing to do with one another! Therefore,
the logical conditional allows implications to be true even when the
hypothesis and the conclusion have no logical connection.

EXAMPLES
Given:
x: Gisele has a math assignment.
y: David owns a car.
Problem: Write x→y as a sentence.
Solution: The conditional x→y represents, "If Gisele has a math
assignment, then David owns a car.

TRY THESE!
Given:
r: 8 is an odd number.
s: 9 is composite.
Problem: 1. What is the truth value of r→s?
2. What is the truth value of s→r?

TRY THESE!
Given:
p: 72 = 49.
q: A rectangle does not have 4 sides.
r: Mario Maurer is a Thai actor.
s: A square is not a quadrilateral.
Problem: Write each conditional below as a sentence. Then indicate
its truth value.
1. p→q 2. q→r
3. p→r 4. q→s
5. r→¬p 6. ¬r →p

Construct a truth table for the statement (m∧∼p)→r
TRY THESE!

For any conditional, there are three related statements, the converse,
the inverse, and the contrapositive.
The original conditional is “if p, then q” p → q
The CONVERSE is “if q, then p” q → p
The INVERSE is “if not p, then not q” ¬ p→ ¬ q
The CONTRAPOSITIVE is “if not q, then not p” ¬ q→¬ p
VARIATIONS ON THE CONDITIONAL
STATEMENT

Consider the conditional “If it is raining, then there are clouds in the sky.” It
seems reasonable to assume that this is true.
Converse:
“If there are clouds in the sky, then it is raining.” This is not always true.
Inverse:
“If it is not raining, then there are no clouds in the sky.” Likewise, this is not
always true.
Contrapositive:
“If there are no clouds in the sky, then it is not raining.” This statement is true,
and is equivalent to the original conditional.
STATEMENT

By the following table, we can identify the values of Converse, Contrapositive,
and Inverse:
Looking at truth tables, we can see that the original conditional and the
contrapositive are logically equivalent, and that the converse and inverse are
logically equivalent.
STATEMENT

A conditional statement and its contrapositive are logically equivalent.
The converse and inverse of a conditional statement are logically equivalent.
In other words, the original statement and the contrapositive must agree with
each other; they must both be true, or they must both be false. Similarly, the
converse and the inverse must agree with each other; they must both be true, or
they must both be false.
STATEMENT

EXAMPLES:
1. What is the contrapositive, the converse, and the inverse of the conditional
statement “The home team wins whenever it is raining.”?
Solution:
Because “q whenever p” is one way to express conditional statements p → q.
Original sentence: “If it is raining, then the home team wins”.
Contrapositive:
“If the home team does not win, then it is not raining.”
Converse:
“If the home team wins, then it is raining.”
Inverse:
“If it is not raining, then the home team does not win.”

EXAMPLES:
2. What are contrapositive, the converse, and the inverse of the conditional
statement “If the picture is a triangle, then it has three sides.”?
Solution:
Contrapositive:
“If the picture doesn’t have three sides, then it is not a triangle.”
Converse:
“If the picture has three sides, then it is a triangle.”
Inverse:
“If the picture is not a triangle, then it doesn’t have three sides.”

TRY THESE!
Write the converse, the inverse, and the contrapositive of the
conditional statement
1. If two angles are congruent, then they have the same measure.
2. If a quadrilateral is a rectangle, then it has two pairs of parallel
sides.
3. If Jennifer is alive, then Jennifer eats food.

BICONDITIONAL STATEMENTS
Examine the sentences below.
Given:
p: A polygon is a triangle.
q: A polygon has exactly 3 sides.
Problem: Determine the truth values of this statement: ( p → q )∧(q →
p)
The compound statement (p → q) ∧ (q → p) is a conjunction of two conditional
statements. In the first conditional, p is the hypothesis and q is the conclusion;
in the second conditional, q is the hypothesis and p is the conclusion.

Let's look at a truth table for this compound statement.
p q p → q q → p (p → q) ∧ (q → p)
T T T T T
T F F T F
F T T F F
F F T T T
In the truth table above, when p and q have the same truth values, the
compound statement (p → q) ∧ (q → p) is true. When we combine two
conditional statements this way, we have a BICONDITIONAL.

A biconditional statement is defined to be true whenever both
parts have the same truth value. The biconditional operator is
denoted by a double-headed arrow ↔. The biconditional p ↔ q
represents "p if and only if q," where p is a hypothesis and q is a
conclusion.
A biconditional statement is a logic statement that includes the
phrase, "if and only if," sometimes abbreviated as "iff." The logical
biconditional comes in several different forms:
 p iff q
 p if and only if q
 p↔q

The following is a truth table for biconditional p ↔ q.
p q p ↔ q
T T T
T F F
F T F
F F T
In the truth table above, p ↔ q is true when p and q have the same truth
values, (i.e., when either both are true or both are false.)

EXAMPLES
1. Given:
p: A polygon is a triangle.
q: A polygon has exactly 3 sides.
Problem: What does the statement p ↔ q represent?
Solution:
The statement p ↔ q represents the sentence, "A polygon
is a triangle if and only if it has exactly 3 sides."
Note that in the biconditional above, the hypothesis is: "A polygon is a triangle" and the
conclusion is: "It has exactly 3 sides." It is helpful to think of the biconditional as a conditional
statement that is true in both directions.

EXAMPLES
2. Given:
a: x + 2 = 7
b: x = 5
Problem:
Write a ↔ b as a sentence. Then determine its
truth values a ↔ b.

Solution:
The biconditional a ↔ b represents the sentence:
"x + 2 = 7 if and only if x = 5."
When x = 5, both a and b are true. When x ≠ 5, both a and b are false. A biconditional
statement is defined to be true whenever both parts have the same truth value. Accordingly,
the truth values of a ↔ b are listed in the table below.
a b a ↔ b
T T T
T F F
F T F
F F T

EXAMPLES
3. Given:
p: x + 7 = 11
q: x = 5
Problem: Is this sentence biconditional? "x + 7 = 11 iff x = 5."
Solution:
Let p → q represent "If x + 7 = 11, then x = 5.“
Let q → p represent "If x = 5, then x + 7 = 11.“
The statement p → q is false by the definition of a conditional. The
statement q → p is also false by the same definition. Therefore, the
sentence "x + 7 = 11 iff x = 5" is not biconditional.

TRY THESE!
1. Create a truth table for the statement (a ∨ b)↔∼ c.

TRY THESE!
2. Let p, q, and r represent the following statements:
p: Sam had pizza last night.
q: Chris finished her homework.
r: Pat watched the news this morning.
Give a formula (using appropriate symbols) for each of these statements.
a. Sam had pizza last night if and only if Chris finished her homework.
b. Pat watched the news this morning iff Sam did not have pizza last night.
c. Pat watched the news this morning if and only if Chris finished her homework
and Sam did not have pizza last night.
d. In order for Pat to watch the news this morning, it is necessary and sufficient
that Sam had pizza last night and Chris finished her homework.

TRY THESE!
3. Define the propositional variables as in Problem 2. Express in words the
statements represented by the following formulas:
p: Sam had pizza last night.
q: Chris finished her homework.
r: Pat watched the news this morning.
a. q ↔ r
b. p ↔ (q ∧ r)
c. ~p ↔ (q ∨ r)
d. r ↔ (p ∨ q)

1. Let p: Jupiter is a planet and q: India is an island be any two simple
statements. Write sentences describing each of the following
statements.
a. ¬p
b. p v ¬q
c. ¬p v q
d. p → ¬q
e. p ↔ q

2. Write each of the following sentences in symbolic form using
statement variables p and q .
a. 19 is not a prime number and all the angles of a triangle are
equal.
b. 19 is a prime number or all the angles of a triangle are not equal
c. 19 is a prime number and all the angles of a triangle are equal
d. 19 is not a prime number

3. Determine the truth value of each of the following statements
a. If 6 + 2 = 5 , then the milk is white.
b. China is in Europe or √3 is an integer.
c. It is not true that 5 + 5 = 9 or Earth is a planet.
d. 11 is a prime number and all the sides of a rectangle are equal.

TAUTOLOGIES, CONTRADICTION,
CONTINGENCY
A TAUTOLOGY is a compound statement S that is true for all possible
combinations of truth values of the component statements that are part
of S.
A CONTRADICTION is a compound statement that is false for all possible
combinations of truth values of the component statements that are part
of S.
A CONTINGENCY is a logical proposition that is neither a tautology nor a
contradiction.

Examples:
For statements p and q:
1. Use a truth table to show that (p ∨ ~p) is a tautology.
2. Use a truth table to show that (p ∧ ~p) is a contradiction.
3. Use a truth table to determine if p → (p ∨ p) is a tautology, a
contradiction, or contingency.

TRY THESE!
Construct the truth table for each of the following and determine
whether the compound statement is a tautology, contradiction, or
contingency.
1. 𝑟 ∧ 𝑝 → 𝑞 → 𝑞
2. 𝑝 → 𝑞 → 𝑟
3. 𝑝 → 𝑝 ↔ 𝑟

Construct the truth table for each of the following and determine whether
the compound statement is a tautology, contradiction, or contingency.
1. p ∧ ∼p 6. (p ↔ r) → [∼q → (p ∧ r)]
2. [p ∧ (p → q)] → ∼q 7. [(p → q) ∧ (q → r)] ∧ ( p ∧ ∼r)
3. (p v ~p) ∧ (q ∧ ~q) 8. [p → (q → r)] ∧ (p → r)
4. ∼(p → q) ∨ [∼p ∨ (p ∧ q)] 9. (p ↔ q) v ~ [(~p ∧ q) v (q ∧ ~q)]
5. ~(p v ~~q) ∧ (p ∧ ~q) 10. [( p → q ) ∧ (q → r)]→ ( p → r)

LOGICAL OPERATIONS SUMMARY
𝒑 𝒒 ¬𝒑 𝒑 ∧ 𝒒 𝒑 ∨ 𝒒 𝒑 → 𝒒 𝒑 ↔ 𝒒
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T

Let
p: “Willis is a good teacher.” q: “Jena is a good teacher.”
r: “Willis' students hate math.” s: “Jena's students hate math.”
Express the following in words:
1. 𝒑 ∧ ∼ 𝒓 2. ∼ 𝒑 ∧ ∼ 𝒒
3. 𝒑 ∨ 𝒓 ∧ ∼ 𝒒 4. 𝒓 ∨ ∼ 𝒑 ∧ 𝒒
5. 𝒒 ∨ ∼ 𝒒 6. ∼ 𝒑 ∧ ∼ 𝒔 ∨ 𝒒
7. 𝒓 ∧ ∼ 𝒓 8. ∼ 𝒔 ∨ ∼ 𝒓
9.∼ 𝒒 ∨ 𝒔 10.∼ 𝒑 ∧ 𝒓

Let A =“Aldo is Italian” and B =“Bob is English”.
Formalize the following sentences:
1. “Aldo isn’t Italian”
2. “Aldo is Italian while Bob is English”
3. “If Aldo is Italian then Bob is not English”
4. “Aldo is Italian or if Aldo isn’t Italian then Bob is English”
5. “Either Aldo is Italian and Bob is English, or neither Aldo is Italian
nor Bob is English”

Let’s consider a propositional statement where
A =“Angelo comes to the party”, B =“Bruno comes to the party”,
C =“Carlo comes to the party”, D =“David comes to the party”.
1. “If David comes to the party then Bruno and Carlo come too”
2. “Carlo comes to the party only if Angelo and Bruno do not come”
3. “David comes to the party if and only if Carlo comes and Angelo doesn’t
come”
4. “If David comes to the party, then, if Carlo doesn’t come then Angelo
comes”

Let’s consider a propositional statement where
A =“Angelo comes to the party”, B =“Bruno comes to the party”,
C =“Carlo comes to the party”, D =“David comes to the party”.
5. “Carlo comes to the party provided that David doesn’t come, but, if David
comes, then Bruno doesn’t come”
6. “A necessary condition for Angelo coming to the party, is that, if Bruno and
Carlo aren’t coming, David comes”
7. “Angelo, Bruno and Carlo come to the party if and only if David doesn’t
come, but, if neither Angelo nor Bruno come, then Davide comes only if
Carlo comes”

TITLE LOREM IPSUM DOLOR SIT AMET
2017
Lorem ipsum dolor sit amet
2018
2019

Logic.pptx

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