Conditionals and Biconditionals in Logic
undefined
Conditionals, Biconditionals,
and Definitions
Unit 2
Conditional Statements
▪
Conditional statements
can also be referred to as
“if-then”
statements.
▪
Example: If you are not completely satisfied,
then your money will be refunded.
▪
Every conditional has two parts. They are:
▪
Hypothesis
– The part following
if
▪
Conclusion
– The part following
then
Identifying the Hypothesis and Conclusion
Example 1:
If today is the first day of fall, then the
month is September.
Hypothesis:
______________________________
Conclusion:
______________________________
Example 2:
If y – 3 = 5, then y = 8.
Hypothesis:
______________________________
Conclusion:
______________________________
Writing a Conditional Statement
Write each sentence as a conditional.
Example 1:
A rectangle has four right angles.
_____________________________________________________
Example 2:
A tiger is an animal.
_____________________________________________________
Example 3:
An integer that ends with 0 is divisible by 5.
_____________________________________________________
Truth Value
▪
Conditionals can have a truth value,
which is either
true
or
false
.
▪
To show that a conditional is true,
show that every time the
hypothesis is true, the conclusion is
also true.
▪
To show that a conditional is false,
you need to find only one
counterexample for which the
hypothesis is true and the
conclusion is false.
Examples:
Show that his conditional is false
by finding a counterexample:
▪
If it is February, then there are
only 28 days in the month.
▪
If the name of a state contains
the word
New
, then the state
borders an ocean.
Converse
The
converse
of a conditional switches the hypothesis and the
conclusion.
Conditional:
If p, then q.
Converse:
If q, then p.
Symbolic
p → q
q → p
Example:
Write the converse of the following conditional
statement.
Conditional:
If
two lines intersect to form right angles
,
then
they are perpendicular
.
Converse:
If
two lines are perpendicular
, then
they
intersect to form right angles
.
Converse
OYO Example:
Write the converse of the following
conditional statement.
Conditional:
If two lines are not parallel and do not
intersect, then they are skew.
Converse:
___________________________________
Conditional:
If an angle is a straight line, then its
measure is 180
o
.
Converse:
___________________________________
Inverse
The
inverse
of a conditional is when you make the conditional
statement negative. (You add “not” to the hypothesis and
conclusion)
Conditional:
If p, then q.
Inverse: If ~q, then ~p.
Symbolic
p → q
~q → ~p
Example:
Write the inverse of the following conditional
statement.
Conditional:
If
two lines intersect to form right angles
,
then
they are perpendicular
.
Inverse:
If
two lines do not intersect to form right angles
,
then
they are not perpendicular
.
Contrapositive
The
contrapositive
is when you make the converse statement negative. (You
add “not” to the hypothesis and conclusion)
Converse:
If q, then p.
Contrapositive: If ~q, then ~p.
Symbolic
q → p
~q → ~p
Example:
Write the contrapositive of the following conditional
statement.
Conditional:
If
two lines intersect to form right angles
,
then
they are perpendicular
.
Converse:
If
two lines are perpendicular
,
then
they intersect to form right angles
.
Contrapositive:
If
two lines are not perpendicular
,
then
they do not intersect to form right angles
.
Example
All obtuse angles have measures greater than 90 degrees.
Conditional:
________________________________________________
Converse:
________________________________________________
Inverse:
________________________________________________
Contrapositive:
________________________________________________
Homework:
Practice
Page 2-1A
#1-8
Page 1-1B
#1-9
Biconditionals and
Definitions
Biconditionals
When a conditional and its converse are true, you can
combine them as a true
biconditional
.
This is the statement you get by connecting the conditional
and its converse, writing it more concisely by joining the
two parts with
if and only if
.
Examples
Conditional:
If
two angles have the same measure
,
then
the angles are congruent
.
Converse:
If
two angles are congruent
, then
the
angles have the same measure
.
Are these both true?
Biconditional:
Two angles have the same measure
if and
only if
the angles are congruent
.
Examples
Consider this true conditional statement. Write its
converse. If the converse is also true, combine the
statements as a biconditional.
Conditional:
If three points are collinear, then they lie
on the same plane.
Converse:
____________________________________
Are both true? If so write the biconditional statement.
___________________________________________________
Examples
Write two statements that form each biconditional.
1.
A line bisects a segment if and only if the line intersects
the segment only at its midpoint.
2.
Two lines are parallel if and only if they are coplanar
and do not intersect.
Definitions
▪
A good definition is a statement that can help you
identify or classify an object. A good definition has
several important components.
▪
A good definition uses clearly understood terms.
▪
A good definition is precise. Avoid words such as
large
,
sort of
, and
some
.
▪
A good definition is reversible. That means you can write a
good definition as a true biconditional.
Write a Definition as a Biconditional
Definition:
Perpendicular lines are two lines that intersect to form right angles.
Conditional:
If two lines are perpendicular, then they intersect to form right angles.
Converse:
If two lines intersect to form right angles, then they are
perpendicular.
Biconditional:
Two lines are perpendicular if and only if they intersect to form right
angles.
Write a Definition as a Biconditional
Definition:
A right angle is an angle whose measure is 90
o
.
Conditional:
Converse:
Biconditional:
Conditional statements, also known as if-then statements, play a crucial role in logic. They consist of a hypothesis (following "if") and a conclusion (following "then"). By identifying the hypothesis and conclusion, writing conditional statements, evaluating truth values, and exploring converses, one can grasp the fundamental concepts of conditionals and biconditionals.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. Download presentation by click this link. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
E N D
Presentation Transcript
Unit 2 Conditionals, Biconditionals, and Definitions
Conditional Statements Conditional statements can also be referred to as if-then statements. Example: If you are not completely satisfied, then your money will be refunded. Every conditional has two parts. They are: Hypothesis The part following if Conclusion The part following then
Identifying the Hypothesis and Conclusion Example 1: If today is the first day of fall, then the month is September. Hypothesis: ______________________________ Conclusion: ______________________________ Example 2: If y 3 = 5, then y = 8. Hypothesis: ______________________________ Conclusion: ______________________________
Writing a Conditional Statement Write each sentence as a conditional. Example 1: A rectangle has four right angles. _____________________________________________________ Example 2: A tiger is an animal. _____________________________________________________ Example 3: An integer that ends with 0 is divisible by 5. _____________________________________________________
Truth Value Examples: Conditionals can have a truth value, which is either true or false. To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. Show that his conditional is false by finding a counterexample: If it is February, then there are only 28 days in the month. If the name of a state contains the word New, then the state borders an ocean. To show that a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.
Converse The converse of a conditional switches the hypothesis and the conclusion. Conditional: If p, then q. Symbolic p q Example: Write the converse of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Converse: If q, then p. q p
Converse OYO Example: Write the converse of the following conditional statement. Conditional: If two lines are not parallel and do not intersect, then they are skew. Converse: ___________________________________ Conditional: If an angle is a straight line, then its measure is 180o. Converse: ___________________________________
Inverse The inverse of a conditional is when you make the conditional statement negative. (You add not to the hypothesis and conclusion) Conditional: If p, then q. Symbolic p q Example: Write the inverse of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Inverse: If two lines do not intersect to form right angles, then they are not perpendicular. Inverse: If ~q, then ~p. ~q ~p
Contrapositive The contrapositive is when you make the converse statement negative. (You add not to the hypothesis and conclusion) Converse: If q, then p. Contrapositive: If ~q, then ~p. Symbolic q p Example: Write the contrapositive of the following conditional statement. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Contrapositive: If two lines are not perpendicular, then they do not intersect to form right angles. ~q ~p
Example All obtuse angles have measures greater than 90 degrees. Conditional: ________________________________________________ Converse: ________________________________________________ Inverse: ________________________________________________ Contrapositive: ________________________________________________
Homework: Practice Page 2-1A #1-8 Page 1-1B #1-9
Biconditionals and Definitions
Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. This is the statement you get by connecting the conditional and its converse, writing it more concisely by joining the two parts with if and only if.
Examples Conditional: If two angles have the same measure, then the angles are congruent. Converse: If two angles are congruent, then the angles have the same measure. Are these both true? Biconditional: Two angles have the same measure if and only if the angles are congruent.
Examples Consider this true conditional statement. Write its converse. If the converse is also true, combine the statements as a biconditional. Conditional: If three points are collinear, then they lie on the same plane. Converse: ____________________________________ Are both true? If so write the biconditional statement. ___________________________________________________
Examples Write two statements that form each biconditional. 1. A line bisects a segment if and only if the line intersects the segment only at its midpoint. 2. Two lines are parallel if and only if they are coplanar and do not intersect.
Definitions A good definition is a statement that can help you identify or classify an object. A good definition has several important components. A good definition uses clearly understood terms. A good definition is precise. Avoid words such as large, sort of, and some. A good definition is reversible. That means you can write a good definition as a true biconditional.
Write a Definition as a Biconditional Definition: Perpendicular lines are two lines that intersect to form right angles. Conditional: If two lines are perpendicular, then they intersect to form right angles. Converse: If two lines intersect to form right angles, then they are perpendicular. Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.
Write a Definition as a Biconditional Definition: A right angle is an angle whose measure is 90o. Conditional: Converse: Biconditional:
