Basics On Logic
PROPOSITIONS
A PROPOSITION is a statement that is either true (T) or false (F).
Use letters p, q, r, P, Q, etc. to represent them.
The negation of proposition P is denoted ~P
Examples:
P: Today is Monday
~P: Today is not Monday
Q: There is a quiz next Friday
~Q: There will not be a quiz next Friday
R: 2 is an odd number
~R: 2 is not an odd number (or 2 is an even number)
COMPOUND PROPOSITIONS
P, Q atomic propositions
New propositions
• P or Q
• P and Q
• If P then Q
• P if and only if Q (general form of a definition)
Truth-Value of Compound Propositions
Proposition P or Q is FALSE when both P and Q are false otherwise true
Proposition P and Q is TRUE when both P and Q are true, otherwise
false
Proposition if P then Q is false only when P is true and Q is false (from
a true fact you can not derive a false result). In any other case it is true.
P is called the condition and Q the conclusion. If P then Q is the same
as saying P is sufficient for Q, or also, Q is necessary for P.
P if and only if Q is true when P and Q are true at the same time or P
and Q are false at the same time.
EXAMPLE
Determine the truth-value of
a. JFK was president of the USA in 1959 or 1960
b. Paul is a CS major and has to take biology
and chemistry
c. If 3 is an even number, then 2 is even
Exercise 1
Determine the truth-value of each of the following statements. To
accomplish it identify the atomic statements, determine their true
value, and finally use the connectors involved in the statement to
determine the truth-value of the compound statement
a. I am a student at TAMUCC and I am a science major
b. 3 divides 6 or 3 divides 10
c. If Δ is a square, Δ is a triangle
Exercise 2
In this exercise you are given compound statements and their truevalue. Identify the connector(s) in the statement, the atomic
propositions participating in the compound proposition, and the
possible truth-values of the atomic propositions. Feel free to label the
atomic propositions using letters.
a. Either John wins the lottery or he finds a job (FALSE)
c. Ruth is majoring in math and biology. (FALSE)
e. The equation 2x - 2 = x2 - x has zero, one, or two real solutions. (TRUE)
EXERCISE 3
It was said that a statement of the form if P THEN Q is false only when P is
true and Q is false (from a true fact you can’t derive a false conclusion).
Consider the situation below to convince yourself about this claim.
“The prize for the Texas lotto this week is $50 millions. Smart Janet decided to
buy a ticket and made the following promise to his classmates:
“If I win the lottery, I will take all of you to the Bahamas with all expenses
paid”
Discuss for which of the following scenarios her statement is true (she told
the truth or fulfilled her promise), and for which it is false (she lied or did not
fulfill her promise).
• Janet won the lottery and took all the classmates to the Bahamas
• Janet won the lottery but did not fulfill her promise
• Janet did not win the lottery but took her classmates to the Bahamas
• Janet did not win the lottery and did not take her classmates to the
Bahamas
EXERCISE 4
Determine the truth-value of each of the following
implications
1. If 2 is negative, - 4 is positive
2. If 4 is even, 3 is even
3. If 3 is odd, 4 is odd
EXERCISE 5
For each of the implications below identify the condition and the
conclusion. The truth-value of the implication is given (in parenthesis).
Use the truth-value of the implication to determine the truth-value of
the condition, and the truth-value of the conclusion. Sometimes your
answer may have several choices of truth-values.
1. If the arrow is green, you have to turn left. (FALSE)
2. If today is cloudy then it will rain (TRUE)
3. If Rob is taking calculus I at TAMU-CC, Rob is either a math
major or a CS major (FALSE)
1. The Music Appreciation class is offered only for students
who are majoring in Music and Science (TRUE)
1. If Chechi is from Mars, he is smart (FALSE)
STATEMENTS WITH “SOME”, “ALL”
Statement may refer to a particular object in a set, to some of them or
to all of them. For instance the statement below refer to the set of
students in calculus I.
Jacob is a student in calculus I (Is Jacob enrolled in the calculus I
class?)
Some calculus I are Computer Science majors
How many?
All students in Calculus I are Engineering majors
How many?
When a statement refers to all the objects in a collection of objects (a
set) the statement is said to be a universally quantified statement.
When a statement refers to some of the objects in a collection (at least
one) the statement is said to be an existentially quantified statement.
Universal Quantified Statements
Science students at TAMUCC have to take calculus I.
Other ways to express the same statement are:
–
–
–
–
Any science student at TAMUCC has to take calculus I
All science students at TAMUCC have to take calculus I
Every (each) science student at TAMUCC has to take calculus I
If you are a science student at TAMUCC then you have to take
calculus I.
It is a universally quantified statement for the collection of science
students at TAMUCC. There are two collections (sets) involved in this
discussion. One is the set of all science students at TAMUCC; the other
one is the set of calculus students at TAMUCC. The quantifier is applied
only to the collection of science majors, which is called the universal set
for the statement.
The statement is true only if each science student takes calculus I.
Otherwise it is false.
Venn Diagram
Calculus I students
Science Students
Any science student at TAMUCC takes calculus I
Contrapositive of
If you don’t take calculus I, you are not a science student
Existential Quantified Statements
Some science students at TAMUCC take Greek.
Other ways to express the same statement are:
– There is a science student at TAMUCC who takes Greek
– At least one science student at TAMUCC takes Greek
It is an existentially quantified for the set of science
students. It refers only to some of the science students at
TAMUCC. In this case you are referring to objects in two
sets. One is the set of TAMUCC students; the other set is
the set of students taking Greek.
The statement is true if there is at least one science
student who takes Greek.
Venn Diagram
Greek students
Science Students
Some science students at TAMUCC take Greek
EXERCISE 6
Write the contrapositive for each of the
following implications
• Any basketball player is taller than 5 feet.
• No calculus student is taking the karate class
EXERCISE 7
Indicate whether each of the following
statements is existentially or universally
quantified. Then determine their truth-value.
• Each student in the class wears glasses
• There is a student in the class who is at least 7
ft. tall
• Some students in the class are females
• All students are not classified as freshmen.
Exercise 8
This exercise deals with the negation of “for
some” and “ for all” using Venn Diagrams.
The statement below the diagram on each row
describes the Venn diagram. The second
diagram represents the negation of the first one.
Your task is to write the proposition for the
second Venn diagram and convince yourself that
statement is the negation of the one to its left.
Exercise 9
Draw the Venn Diagrams for each of the statements
below. Then determine the truth-value of each of them.
a. If x is a real number different than zero, it has a
multiplicative inverse.
d. Any integer number is odd.
g. Any science major takes a math course, but some
students taking a math course are not science majors.
COUNTER EXAMPLES
From our previous discussions it follows that the negation of
“Any calculus I student wears glasses”
Is
“ There is at least one calculus student who does not wear glasses”.
To guarantee that “Any calculus I student wears glasses” is
false, it ha to be guaranteed that “ There is at least one
calculus student who does not wear glasses” is true.
To do this it is enough to show one student in the class that
does not wear glasses, for instance Lisa. Lisa does not wear
glasses is a counterexample to show that “Any calculus I
student wears glasses” is false.
EXERCISE 10
Give a counterexample to show that each of the statements below is
false.
a. Any TAMUCC is from Texas
b. All numbers which are multiples of 3 are multiples of 6
c. All calculus I students at TAMUCC are science majors
d. Any NBA player is at least 6 feet tall.