Transcript p ~q - NEHSMath

Section 2-2: Biconditional and Definitions
TPI 32C: Use inductive and deductive reasoning to make conjectures
Objectives:
• Write the inverse and contrapositive of conditional statements
• Write Biconditionals and recognize good definitions
Conditional Statements and Converses
Statement
Example
Symboli
c
You read as
Conditional
If an angle is a straight angle,
then its measure is 180º.
pq
If p, then q.
Converse
If the measure of an angle is
180º, then it is a straight
angle.
qp
If q then p.
Forms of a Conditional Statement
Converse
Inverse
Contrapositive
Biconditional
Symbolic Negation (~p  ~q)
• Negation of a statement has the opposite truth value.
Statement:
ABC is an obtuse angle.
Negation:
ABC is not an obtuse angle.
Statement:
Lines m and n are not
perpendicular
Negation:
Lines m and n are
perpendicular.
Form of a Conditional Statement
Symbol ~ is used to indicate the word “NOT”
(~p~q)
If not p, then not q.
States the opposite of both the hypothesis and conclusion.
Conditional:
pq : If two angles are vertical, then they are congruent.
Inverse:
~p~q: If two angles are not vertical, then they are not
congruent.
Inverse
• Inverse of a conditional negates BOTH the hypothesis and conclusion.
Conditional
If a figure is a square, then it is a rectangle.
NEGATE BOTH
Inverse
If a figure is NOT a square, then it is NOT a rectangle.
Form of a Conditional Statement
(~q~p)
If not q, then not p.
Switch the hypothesis and conclusion & state their opposites.
(~q~p) (Do Converse and Inverse)
Conditional:
pq : If two angles are vertical, then they are congruent.
Contrapositive:
~q~p: If two angles are not congruent, then they are not
vertical.
Contrapositive
• Contrapositive switches hypothesis and conclusion AND
negates both.
• A conditional and its contrapositive are equivalent. They
have the same truth value).
Conditional
If a figure is a square, then it is a rectangle.
SWITCH AND NEGATE BOTH
Contrapositive
If a figure is NOT a rectangle, then it is NOT a square.
Lewis Carroll, the author of Alice's
Adventures in Wonderland and
Through the Looking Glass, was
actually a mathematics
teacher. As a hobby, Carroll wrote
stories that contain amusing
examples of logic. His works
reflect his passion for mathematics
Lewis Carroll’s “Alice in Wonderland” quote:
"You might just as well say," added the Dormouse, who
seemed to be talking in his sleep, "that 'I breathe when I
sleep' is the same thing as 'I sleep when I breathe'!"
Translate into a conditional:
If I am sleeping, then I am breathing.
Inverse of a conditional:
If I am not sleeping, then I am not
breathing.
Contrapositive of a conditional:
If I am not breathing, then I am not
sleeping.
Form of a Conditional Statement
pq
• Write a bi-conditional only if BOTH the conditional and the
converse are TRUE.
• Connect the conditional & its converse with the word “and”
• Write by joining the two parts of each conditional with the
phrase “if and only if” of “iff” for shorthand.
• Symbolically: p  q
Bi-conditional Statements
Conditional Statement:
If two angles same measure, then the angles are congruent.
Converse:
If two angles are congruent, then they have the same
measure.
Both statements are true, so….
…you can write a Biconditional statement:
Two angles have the same measure if and only if the
angles are congruent.
Write a Bi-conditional Statement
Consider the following true conditional statement. Write its
converse. If the converse is also true, combine the
statements as a biconditional.
Conditional:
If x = 5, then x + 15 = 20.
Converse:
If x + 15 = 20, then x = 5.
Since both the conditional and its converse are true, you
can combine them in a true biconditional using the phrase
if and only if.
Biconditional:
x = 5 if and only if x + 15 = 20.
Separate a Biconditional
• Write a biconditional as two conditionals that are
converses of each other.
Consider the biconditional statement:
A number is divisible by 3 if and only if the sum of its digits
is divisible by 3.
Statement 1:
If a number is divisible by 3, then the sum of its digits is
divisible by 3.
Statement 2:
If the sum of a numbers digits is divisible by 3, then the
number is divisible by 3.
Separate a Biconditional
Write the two statements that form this biconditional.
Biconditional:
Lines are skew if and only if they are noncoplanar.
Conditional:
If lines are skew, then they are noncoplanar.
Converse:
If lines are noncoplanar, then they are skew.
Writing Definitions as Biconditionals
• Good Definitions:
 Help identify or classify an object
 Uses clearly understood terms
 Is precise avoiding words such as sort of and some
 Is reversible, meaning you can write a good definition as a
biconditional (both conditional and converse are true)
Show definition of perpendicular lines is reversible
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.
Since both are true converses of each other, the definition can be
written as a true biconditional:
“Two lines are perpendicular iff they intersect to form right angles.”
Writing Definitions as Biconditionals
Show that the definition of triangle is reversible. Then write
it as a true biconditional.
Steps
1. Write the conditional
2. Write the converse
3. Determine if both statements are true
4. If true, combine to form a biconditional.
Definition: A triangle is a polygon with exactly three sides.
Conditional:
If a polygon is a triangle, then it has exactly three sides.
Converse:
If a polygon has exactly three sides, then it is a triangle.
Biconditional:
A polygon is a triangle if and only if it has exactly three sides.
Writing Definitions as Biconditionals
Is the following statement a good definition? Explain.
An apple is a fruit that contains seeds.
Conditional: If a fruit is an apple then if contains seeds.
Converse: If a fruit contains seed then it is an apple.
There are many other fruits containing seeds that are not
apples, such as lemons and peaches. These are
counterexamples, so the reverse of the statement is false.
The original statement is not a good definition because the
statement is not reversible.
Statement
Example
Symbolic
You read as
Conditional
If an angle is a straight angle,
then its measure is 180º.
pq
If p, then q.
Converse
If the measure of an angle is
180º, then it is a straight angle
qp
If q then p.
Inverse
If an angle is not a straight
angle, then its measure is not
180.
~p  ~q
If not p, then
not q
Contrapositive
If an angle does not measure
180, then the angle is not a
straight angle.
~q  ~p
If not q, then
not p.
Biconditional
An angle is a straight angle if
and only if its measure is 180º.
pq
p if and only if
q.
P iff q