Mathematical Logic
The Converse, the inverse, and the
Contrapositive
The conditional statement, (antecedent) (consequent) has the
following terms.
a. If p, then q g. q provided that p
b. p implies q h. q is a necessary condition for p
c. P if and only if q i. p is a sufficient condition for q
d. Not p or q
e. Every p is q j. p yields q
f. q, if p k. q follows from p.
Example 1
Consider the following statements,
p: x is a prime number
q: x is odd
Formalize the following statements.
a. x being prime is a sufficient condition for being odd.
b. x being odd is a necessary condition for being prime.
Solution:
Example 2
Write each statement in the form “if p, then q.”
a. I will be able to tour abroad next year provided that I have my
passport renewed.
b. Every triangle that is not a right triangle is an oblique triangle.
Solution:
a. If I have my passport renewed, then I will be able to tour abroad
next year.
b. If a triangle is not a right triangle, then it is an oblique triangle.
Derived Forms of a Conditional Statement
There are three ways to restate a conditional statement – the
converse, the inverse and the contrapositive. However, restating the
statement into one of these forms may change the meaning of the
original statement. The conditional statement, , may be restated in
the following forms
Converse form:
Inverse form:
Contrapositive:
Example 3
Write the converse, inverse, and contrapositive of the given sentence.
“She is allowed to join the volleyball team, only if she knows how to receive
the ball.”
Solution:
If she is allowed to join the volleyball team, then she knows how to
receive the ball.
Converse ( ): If she knows how to receive the ball, then she is allowed
to join the volleyball team
Inverse ( ): If she is not allowed to join the volleyball team, then she
does not know how to receive the ball.
Contrapositive ( ): If she does not know how to receive the ball, then
she is not allowed to join the volleyball team
Example 4
Write the converse, inverse, and contrapositive of the given sentence.
“Every rectangle is a parallelogram.”
Solution:
If it is a rectangle then it is a parallelogram.
Converse ( ): If it is a parallelogram then it is a rectangle.
Inverse ( ): If it is not a rectangle then it is not a parallelogram.
Contrapositive ( ): If it is not a parallelogram then it is not a rectangle.
“If 10 is an even number then it is divisible by 2”
Compound statements and their truth tables
Definition: Compound Statements are complex statements built up on
two simple statements by using connectives.
In joining two statements, parenthesis is used to enclose a single
statement. The truth table of a compound statements involving two or
more statements can be constructed from the truth tables of each of the
single statements.
Example 5: Construct the truth table for the compound statement
Solution:
𝑝 𝑞 (𝑝 ∨ 𝑞) ∼𝑝 𝑝∨𝑞 ∧∼𝑝
𝑇 𝑇 𝑇 𝐹 𝐹
𝑇 𝐹 𝑇 𝐹 𝐹
𝐹 𝑇 𝑇 𝑇 𝑇
𝐹 𝐹 𝐹 𝑇 𝐹
Example 6: Construct the truth table for the compound statement
Solution:
𝑝 𝑞 ∼𝑝 ∼𝑞 ∼𝑞→𝑝 ∼ 𝑝 ∨ (∼ 𝑞
→ 𝑝)
𝑇 𝑇 𝐹 𝐹 𝑇 𝑇
𝑇 𝐹 𝐹 𝑇 𝑇 𝑇
𝐹 𝑇 𝑇 𝐹 𝑇 𝑇
𝐹 𝐹 𝑇 𝑇 𝐹 𝑇
Definition:
If the truth value of a compound statement is always true
regardless of the truth values of each of the component statements then
the statement is said to be a tautology.
Definition:
If the truth value of a compound statement is always false
regardless of the truth values of each of the component statements then
the statement is said to be a contradiction.
Logical Equivalences
Definition:
Two mathematical statements are logically equivalent if the final
output of their truth tables are exactly the same. For example, the
statement is not logically equivalent to but it is logically
equivalent to
If are compound statements, then are logically
equivalent if and only if is a tautology.
Example 7
Verify if the statements and are logically
equivalent. What is the logical content equivalent “If the price is right,
then I will accept the job offer.”?
Solution:
𝑝 𝑞 ∼𝑝 𝒑→𝒒 ∼𝒑∨𝒒
𝑇 𝑇 𝐹 𝑻 𝑻
𝑇 𝐹 𝐹 𝑭 𝑭
𝐹 𝑇 𝑇 𝑻 𝑻
𝐹 𝐹 𝑇 𝑻 𝑻
Since the last two columns are identical, the given statements are
logically equivalent.
Example 7 (continuation)
Using the results, the logical content equivalent of “If the price is right,
then I will accept the job offer.” is “The price is not right or I will accept
the job offer.”
De Morgan’s Law
Let and be any propositional statements. Then,
Example 8: Prove
Proof:
)
.
Quantifiers
Special words like “all”, “any”, “every”, and “some” are called
quantifiers. They can be used to analyze mathematical sentences or may
be used to define mathematical terms. Quantifiers are categorized into
universal quantifiers and existential quantifiers.
Universal Quantifiers
Universal quantifiers such as “all” and “every” are used to denote
that all elements in the set satisfy the given property.
Examples:
a. All professors handling professional courses are all licensed
engineers.
b. Each of the students attending the field trip must have the waiver
signed by the parents.
c. Every right triangle has an angle that measures 90 degrees.
Existential Quantifiers
Existential quantifiers such as “some” and “there exists” are used to
denote that one or more elements of a set satisfy a given property.
Examples:
a. Some parents now fear vaccines.
b. There exist a relationship between the mathematical language and
the ordinary language.
c. There is a number whose square is 9.
Notation for Quantifiers
If S is a set an is a statement about the element , then the notation
means that “for all x in S, P of x is true” or “P(x) is true for every x in
set S.”
The notation
means that “there exist x in S, for which P of x is true” or “There exist at
least one element x of S for which P(x) is true.”
Example 8
Use quantifiers to define odd and even numbers.
Solution:
a. An integer x is even, if there exist an integer y such that . In
symbols
b. An integer x is odd if there exist an integer y such that In
symbols,
Activity!
Translate the following logic symbols into words.
M: Mark is English
L: Lem is German
Write the converse, inverse and contrapositive of the given statement.
“If yesterday is not Wednesday, then tomorrow is not Friday. “
