Logic and Set Theory

1. ELEMENTARY
LOGIC
Davao Oriental State University
Graduate School Department

2. Logic Statements
Every language contains different types of sentences,
such as statements, questions, and commands. For
instance,
• “Is the test today?” is a question.
• “Go get the newspaper” is a command.
• “This is a nice car” is an opinion.
• “Manila is the capital of Philippines” is a statement of
fact.

3. Statement
A statement is a declarative sentence that is either true or
false, but not both true and false.
• It may not be necessary to determine whether a sentence is
true to determine whether it is a statement. For instance,
consider the following sentence.
American Shaun White won an Olympic gold medal in speed
skating.
• You may not know if the sentence is true, but you do know that
the sentence is either true or it is false, and that it is not both
true and false. Thus, you know that the sentence is a statement.

4. Let’s try!
Determine whether each sentence is a statement.
a. Lupon is a municipality in Davao Oriental
b. How are you?
c. 99
+ 2 is a prime number.
d. 𝑥 + 1 = 5.
a. Lupon is a municipality in Davao Oriental, so the sentence is true and it is a statement.
b. The sentence is a question; it is not a declarative sentence. Thus it is not a statement.
c. You may not know whether 99 + 2 is a prime number; however, you do know that it is a whole
number larger than 1, so it is either a prime number or it is not a prime number. The sentence is
either true or it is false, and it is not both true and false, so it is a statement.
d. 𝑥 + 1 = 5 is a statement. It is known as an open statement. It is true for 𝑥 = 4, and it is false for
any other values of 𝑥. For any given value of 𝑥, it is true or false but not both.

5. Simple Statements and Compound
Statements
A simple statement is a statement that conveys a single
idea. A compound statement is a statement that
conveys two or more ideas.
Connecting simple statements with words and phrases such as
and, or, if . . . then, and if and only if creates a compound
statement.
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Example:
“I will attend the meeting or I will go to school.” is a compound statement.

6. Logic Connectives and Symbols

7. Let’s try!
Consider the following simple statements.
p: Today is Friday.
q: It is raining.
r: I am going to a movie.
s: I am not going to the basketball game.
Write the following compound statements in symbolic form.
a. Today is Friday and it is raining.
b. It is not raining and I am going to a movie.
c. I am going to the basketball game or I am going to a movie.
d. If it is raining, then I am not going to the basketball game.
𝒑 ∧ 𝒒
~𝒒 ∧ 𝒓
~𝒔 ∨ 𝒓
𝒒 → 𝒔

8. Let’s try!
Consider the following simple statements.
p: The game will be played in Araneta.
q: The game will be shown on ABS.
r: The game will not be shown in GMA.
s: The Raptors are favored to win.
Write the following symbolic statements in words.
a. 𝒒 ∧ 𝒑
b. ~𝒓 ∧ 𝒔
c. 𝒔 ↔ ~𝒑
The game will be shown on ABS and the game will be played in Araneta.
The game will be shown on GMA and the Raptors are favored to win.
The Raptors are favored to win if and only if the game will not be played in Araneta.

9. Compound Statements and Grouping
Symbols
If a compound statement is written in symbolic form, then parentheses are used
to indicate which simple statements are grouped together.

10. Compound Statements and Grouping
Symbols
If a compound statement is written as an English sentence, then a comma is used to
indicate which simple statements are grouped together. Statements on the same side
of a comma are grouped together

11. Let’s try!
Let p, q, and r represent the following.
p: You get a promotion.
q: You complete the training.
r: You will receive a bonus.
a. Write (𝑝 ∧ 𝑞) → 𝑟 as an English sentence.
If you get a promotion and you complete the training, then you will receive a
bonus.
b. Write “If you do not complete the training, then you will not get a
promotion and you will not receive a bonus.” in symbolic form.
~𝒒 → (~𝒑 ∧ ~𝒓)

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

13. Truth Value and Truth Tables
The negation of the statement “Today is Friday.” is
the statement “Today is not Friday.”
In symbolic logic, the tilde symbol ~ is used to
denote the negation of a statement. If a statement
p is true, its negation ~p is false, and if a statement
p is false, its negation ~p is true.
The negation of the negation of a statement is the
original statement. Thus ~(~p) can be replaced by
p in any statement.

14. Truth Value of a Conjunction
The conjunction 𝑝 ∧ 𝑞 is true if and only if both 𝑝 and 𝑞 are true.

15. Truth Value of a Disjunction
The disjunction 𝑝 ∨ 𝑞 is true if and only if 𝑝 is true, 𝑞 is true, or
both 𝑝 and 𝑞 are true.

16. Truth Value and Truth Tables
Construct a truth table for ~ ~𝑝 ∨ 𝑞 ∨ 𝑞.
𝑝 𝑞
T T
T F
F T
F F
~𝑝
F
F
T
T
~𝑝 ∨ 𝑞
T
F
T
T
~(~𝑝 ∨ 𝑞)
F
T
F
F
~(~𝑝 ∨ 𝑞) ∨ 𝑞
T
T
T
F

17. Truth Value and Truth Tables
Construct a truth table for 𝑝 ∧ 𝑞 ∧ (~𝑟 ∨ 𝑞).
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 ∧ 𝑞
T
T
F
F
F
F
F
F
~𝑟
F
T
F
T
F
T
F
T
~𝑟 ∨ 𝑞
T
T
F
T
T
T
F
T
𝑝 ∧ 𝑞 ∧ (~𝑟 ∨ 𝑞)
T
T
F
F
F
F
F
F

18. Truth Tables (via Order of Precedence)
Construct a truth table for 𝑝 ∨ [~ 𝑝 ∧ ~𝑞 ].

19. Equivalent Statements
Two statements are equivalent if they both have
the same truth value for all possible truth values of
their simple statements. Equivalent statements
have identical truth values in the final columns of
their truth tables. The notation 𝑝 ≡ 𝑞 is used to
indicate that the statements 𝑝 and 𝑞 are
equivalent.

20. Equivalent Statements
Show that ~(𝑝 ∨ ~𝑞) and ~𝑝 ∧ 𝑞 are equivalent statements.

21. De Morgan’s Law
For any statements p and q
De Morgan’s law can be used to restate certain English sentence
in its equivalent form

22. De Morgan’s Law
Use one of De Morgan’s laws to restate the following sentence in
an equivalent form.
It is not true that, I graduated or I got a job.
Let p represent the statement “I graduated” and let q represent
the statement “I got a job”. By ~ 𝑝 ∨ 𝑞 ≡ ~𝑝 ∧ ~𝑞, we have,
I did not graduate and I did not get a job

23. Tautologies and Self-Contradictions
A tautology is a statement that is always
true. A self-contradiction is a statement that
is always false.

24. Tautologies and Self-Contradictions
Show that 𝑝 ∨ (~𝑝 ∨ 𝑞) is a tautology.

25. Tautologies and Self-Contradictions
Show that 𝑝 ∧ (~𝑝 ∧ 𝑞) is a self-contradiction.

26. Conditional Statements
Conditional statements can be written in 𝑖𝑓 𝑝, 𝑡ℎ𝑒𝑛 𝑞 form or
in 𝑖𝑓 𝑝, 𝑞 form. For instance, all of the following are
conditional statements.
• If we order pizza, then we can have it delivered.
• If you go to the movie, you will not be able to meet us for dinner.
• If n is a prime number greater than 2, then n is an odd number.
In any conditional statement represented by “If p, then q” or
by “If p, q,” the p statement is called the antecedent and the q
statement is called the consequent.
The conditional statement, “If p, then q,” can be written using the arrow notation 𝑝 → 𝑞. The
arrow notation is read as “if p, then q” or as “p implies q.”
Institute of Computng and Engineering
Mathematics Department | Jerd M. Dela Gente, MSc.

27. Conditional Statements
Identify the antecedent and consequent in the following
statements.
a. If our school was this nice, I would go there more than once a week.
b. If you don’t stop and look around once in a while, you could miss it.
c. If you strike me down, I shall become more powerful than you can
possibly imagine.
a. Antecedent: our school was this nice
Consequent: I would go there more than once a week
b. Antecedent: you don’t stop and look around once in a while
Consequent: you could miss it
c. Antecedent: you strike me down
Consequent: I shall become more powerful than you can possibly imagine
Institute of Computing and Engineering
Mathematics Department | Jerd M. Dela Gente, MSc.

28. Truth Value and Truth Table for 𝒑 → 𝒒
The conditional 𝑝 → 𝑞 is false if p is true and q is false. It is true in
all other cases.

29. Truth Value and Truth Table for 𝒑 → 𝒒
Determine the truth value of each of the following.
a. If 2 is an integer, then 2 is a rational number.
b. If 3 is a negative number, then 5>7.
c. If 5>3, then 2+7=4.
a. Because the consequent is true, this is a true statement.
b. Because the antecedent is false, this is a true statement.
c. Because the antecedent is true and the consequent is false, this is a false statement.

30. Truth Value and Truth Table for 𝒑 → 𝒒
Construct a truth table for 𝑝 ∧ 𝑞 ∨ ~𝑝 → ~𝑝.

31. An Equivalent Form of the Conditional
The conditional 𝑝 → 𝑞 is equivalent to the disjunction ~𝑝 ∨ 𝑞
𝑝 → 𝑞 ≡ ~𝑝 ∨ 𝑞
Example:
If I could play the guitar, I would join the band.
I cannot play the guitar or I would join the band.

32. Negation of the Conditional
Because 𝑝 → 𝑞 ≡ ~𝑝 ∨ 𝑞, an equivalent form of ~(𝑝 → 𝑞) is given
by ~(~𝑝 ∨ 𝑞), which, by one of De Morgan’s laws, can be
expressed as the conjunction 𝑝 ∧ ~𝑞.
Example:
𝑝 → 𝑞 :If they pay me the money, I will sign the contract.
~ 𝑝 → 𝑞 : They paid me the money and I did not sign the
contract.
~ 𝑝 → 𝑞 ≡ 𝑝 ∧ ~𝑞

33. The Biconditional
The statement (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)is called a biconditional and is
denoted by 𝑝 ↔ 𝑞, which is read as “p if and only if q.”
𝑝 ↔ 𝑞 ≡ (𝑝 → 𝑞) ∧ (𝑞 → 𝑝)

34. Truth Value and Truth Table for 𝒑 ↔ 𝒒
The biconditional 𝑝 ↔ 𝑞 is true only when p and q have the same
truth value.

35. Truth Value and Truth Table for 𝒑 ↔ 𝒒
State whether each biconditional is true or false.
a. 𝑥 + 4 = 7 if and only if 𝑥 = 3.
b. 𝑥2 = 36 if and only if 𝑥 = 6.
a. Both equations are true when 𝑥 = 3, and both are false when 𝑥 ≠ 3. Both equations
have the same truth value for any value of x, so this is a true statement.
b. If 𝑥 = −6, the first equation is true and the second equation is false. Thus this is a false
statement.

36. Symbolic Arguments
An argument consists of a set of statements called
premises and another statement called the conclusion.
An argument is valid if the conclusion is true whenever all
the premises are assumed to be true. An argument is
invalid if it is not a valid argument.

37. Example:
If Aristotle was human, then Aristotle was mortal. Aristotle was human.
Therefore, Aristotle was mortal.
The argument has two premises and 1 conclusion:
First premise: If Aristotle was human, then Aristotle was mortal.
Second premise: Aristotle was human. .
Conclusion: Therefore, Aristotle was mortal.
Let ℎ represent the statement “Aristotle was human” and let 𝑚 represent the statement “Aristotle was mortal” The
argument in symbolic form is,
ℎ → 𝑚
ℎ
∴ 𝑚
Symbolic Arguments
The three dots are a symbol for “therefore.”

38. Example:
The fish is fresh or I will not order it. The fish is fresh. Therefore I will
order it.
Let 𝑓 represent the statement “The fish is fresh” and let 𝑜 represent the statement “I will
order it” The argument in symbolic form is,
𝑓 ∨ ~𝑜
𝑓
∴ 𝑜
Symbolic Arguments

39. Arguments and Truth Table
Validity of the Aristotle example
If the conclusion is true in every row
of the truth table in which all the
premises are true, the argument is
valid. If the conclusion is false in any
row in which all of the premises are
true, the argument is invalid.
Row 1 is the only row in which all the premises are true, so it is the only row that we examine. Because
the conclusion is true in row 1, the argument is valid.

40. Arguments and Truth Table
Determine whether the following argument is valid or invalid.
If it rains, then the game will not be played. It is not raining.
Therefore, the game will be played.
Because the conclusion in row 4 is false and the premises are both true, the argument
is invalid.

41. Arguments and Truth Table
Determine whether the following argument is valid or invalid.
Because the conclusion is true in the rows 2, 6, 7 and 8, the argument is valid.

42. Standard Forms of Valid Arguments

43. Standard Forms of Invalid Arguments

44. Validity of argument examples
Use a standard form to determine whether the following
argument is valid or invalid.
This symbolic form matches the standard form known as
disjunctive syllogism. Thus the argument is valid.

45. Validity of argument examples
Consider an argument with the following symbolic form.
This sequence of valid arguments shows that t is a valid conclusion for the original argument.

46. Validity of argument examples
Determine whether the following argument is valid.
If the movie was directed by Steven Spielberg (𝑠), then I want to see it (𝑤). The movie’s
production costs must exceed $50 million (𝑐) or I do not want to see it. The movie’s
production costs were less than $50 million. Therefore, the movie was not directed by
Steven Spielberg.
This sequence of valid arguments shows that t is a valid conclusion for the original argument.