Download Conditional Statements, Biconditionals, and Deductive Reasoning

Conditional Statements,
Biconditionals, and Deductive
Reasoning
Part 2
Conditional Statements
Conditional Statements
• A conditional is an If – then statement
– p  q (read as if p then q or p implies q)
• The Hypothesis is the part p following if
• The Conclusion is the part q following then.
Identifying the Hypothesis and
Conclusion
• What is the hypothesis and conclusion of the
conditional?
– If an animal is a robin, then the animal is a bird
• H – An animal is a robin
• C – The animal is a bird
– If an angle measures 130, then the angle is obtuse
• H – An angle measures 130
• C – The angle is obtuse
Writing a Conditional
• Write the following statement as a conditional
– Vertical angles share a vertex
• Step 1: Identify the Hypothesis and Conclusion
– H: Vertical Angles
– C: Share a vertex
• Step 2: Write the Conditional
– If two angles are vertical, then they share a common vertex
– You Try: How can you write “Dolphins are
mammals” as a conditional?
• If an animal is a dolphin, then it is a mammal
Truth Value
• The truth value of a conditional is either true
or false.
• To show a conditional is true, show that every
time the hypothesis is true, the conclusion is
also true
• To show a conditional is false find one counter
example for which the hypothesis is true and
the conclusion is false
Finding the Truth Value of a
Conditional
• Is this conditional true or false, if it is false find a
counter example.
– If a women is Hungarian, then she is European.
• This is True!
– If a number is divisible by 3, then it is odd.
• This is false, the number 12 is divisible by three and not odd.
– If a month has 28 days than it is February
• This false, January has 28 days
– If two angles form a linear pair, then they are
supplementary
• True!
Negation
• The negation of a statement p is the opposite
of that statement, the symbol is ~p and is read
“not p”
– Example:
• The negations of the statement “the sky is blue” is “the
sky is not blue”
• You use the negation to write statements
related to a condition
Related Conditional Statements
Statement
How To Write
Example
Symbol
How to read
Conditional
Use the given
hypothesis and
conclusion
If m<A = 15, then
<A is acute
p q
If p, then q
Converse
Exchange the
hypothesis and
conclusion
If <A is acute,
then m<A = 15
q p
If q, then p
Inverse
Negate both the
hypothesis and
conclusion from
the conditional
If m<A ≠ 15, then
<A is not acute
~p  ~ q
If not p, then
not q
Contrapositiv Negate both the
e
hypothesis and
conclusion from
the converse
If <A is not acute,
then m<A ≠ 15
~q  ~p
If not q, then
not p
Truth Value
Statement
Example
Truth Value
Conditional
If m<A = 15, then <A is
acute
True
Converse
If <A is acute, then m<A =
15
False
Inverse
If m<A ≠ 15, then <A is not
acute
False
Contrapositive
If <A is not acute, then
m<A ≠ 15
True
Equivalent Statements have the same truth value,
the conditional and contrapositive are equivalent,
and are the converse and inverse statements.
You Try
• Write the Converse, Inverse and Contrapositive
statements
– IF a vegetable is a carrot, then it contains beta carotene
– Converse
• If a vegetable contains beta carotene then it is a carrot
– False (Spinach has Beta Carotene)
– Inverse
• If a vegetable is not a carrot then it does not contain beta carotene
– False
– Contrapositive
• If a vegetable does not contain beta carotene then it is not a carrot
– True!
Part 3
Biconditionals
Biconditional
• A single true statement that combines a true
conditional and its true converse, you can
write a biconditional by joining the two parts
of each conditional with the phrase if and only
if
– Symbol:
Writing a Biconditional
• To write a biconditional first determine if the
what is the converse of the following true
conditional. If the converse is true then write a
biconditional statement
– Conditional: If the sum of the measure of two angles
is 180, then the two angles are supplementary
– Converse: If two angles are supplementary, then the
sum of the measures of the two angles is 180
• Biconditional:
– Two angles are supplementary if and only if the sum
of the measures of the two angles is 180
You Try
• What is the converse of the following conditional,
if the converse is true write a biconditional
statement
• If two angles have equal measures, then the
angles are congruent
– Converse: If angles are congruent, then they have
equal measures
• Biconditional
– Two angles have equal measures if and only if they are
congruent
Identifying the conditionals in a
Biconditional
• What are the two statements that form a
biconditional
– A ray is an angle bisector if and only if it divides and
angle into two congruent angles
• Find p and q
• P – A ray is an angle bisector
• Q – A ray divides an angle into two congruent angles
– Conditional: If a ray is an angle bisector, then it divides
the angle into two congruent angles
– Converse: If a ray divides and angle into two
congruent angles, then it is an angle bisector
You Try!
• What are the two conditionals that form this
biconditional?
– Two numbers are reciprocals if and only if their
product is one.
• Conditional: If two numbers are reciprocals,
then their product is one
• Converse: If two numbers product is one, then
they are reciprocals.
Part 4
Deductive Reasoning
Deductive Reasoning
• Also called logical reasoning, is the process of
reasoning logically from given statements or
facts to a conclusion
Law of Detachment
• If the hypothesis of a true conditional is true,
then the conclusion is true
• If p then q is true and p is true, then q is true
Using the law of detachment
• What can you conclude from the given true
statements?
– If a student gets an A on a final exam, then the
student will pass the course. Felicia got an A on
her history Final
• Felicia will pass the course
– If a ray divided an angle into two congruent
angles, then the ray is an angle bisector. Ray RS
divides <ARB so that <ARS ≅ <SRB
• Ray RS is an angle bisector
More Examples
– If two angles are adjacent, then they share a common
vertex. <1 and <2 share a common vertex.
• Since the second statement does not match the hypothesis
then we can not conclude anything
– If there is lightning, then it is not safe to be out in the
open. Marla sees lightning from the soccer field.
• It is not safe to be out in the open
– If a figure is a square, then its sides have equal
lengths, figure ABCD has sides of equal length.
• We can not conclude this is a square because out statement
matched the conclusion not the hypothesis
Law of Syllogism
• Allows you to state a conclusion from two true
conditional statements when the conclusion
of one statement is the hypothesis of another
statement
– If p  q is true
– And q  r is true
– Then p  r is true
Using the law of Syllogism
• What can you conclude from the given
information?
• If a figure is a square, then the figure is a
rectangle. If a figure is a rectangle than the figure
has four sides
– If a figure is a square then it has four sides
• If you do gymnastics, then you are flexible. If you
do ballet then you are flexible.
– Each conclusion is the same so we can not use the law
of syllogism and can conclude nothing
More Examples
• If a whole number ends in 0, then it is divisible by
10. If a whole number is divisible by 10, than it is
divisible by 5.
– If a whole number ends in zero is it divisible by 5
• If Ray AB and Ray AD are opposite rays, then the
two rays form a straight angle. If two rays are
opposite rays, then the two rays form a straight
angle.
– The hypothesis and conclusion matches so we can
make no further conclusions
The END!