Download Biconditional Statements

Unit 3
Section 1
Logical
Statements
Conditional Statements
Suppose p and q are statements.
Put them together in the form
If p then q
conditional statement When a
“____________”.
We call this a __________________.
conditional statement is written in this “if-then” form the “if” part is
antecedent which means “___________”.
to go before The “then”
called the __________,
consequent which means
part is called the ____________,
an event to follow another.
“________________________”.
Underline the antecedent with one line and the consequent with two
lines, in each of these conditional statements.
If two lines intersect, then their intersection is exactly one point.
If two planes intersect then their intersection is a line.
If an animal is a tiger, then it has stripes.
Try writing these as if-then statements. In your new
statement, underline the antecedent with one line and
underline the consequent with two lines.
A prime number has exactly two divisors.
If a number is prime then it has exactly two divisors.
Seals swim.
If an animal is a seal then it swims.
All birds have feathers.
If an animal is a bird then it has feathers.
3n is odd if n is odd.
If n is odd then 3n is odd.
3n is odd only if n is odd.
If 3n is odd then n is odd.
When is a conditional statement false?
For the next three statements, how would you show that the statement
is false?
If you live in Kansas, then you live in Leavenworth.
Governor Brownback lives in
Kansas, but not in Leavenworth.
If the product of two numbers is positive, then the two numbers
must both be positive.
(-2)(-8) = 16
counter example The
An example like this is called a _________________.
antecedent is true, but the ________________
_____________
is false in a
consequent
counterexample.
Rewrite these false statements as if-then statements, and
find or draw counterexamples to each one.
All musicians are guitar players.
If a person is a musician, then he/she is a guitar
Piano player
player.
The sum of two even numbers is odd.
If two numbers are even then their sum is odd.
2 + 4 = 6 EVEN
Two rays that have the same endpoint are always opposite rays.
If two rays have the same endpoint then
they are opposite rays.
A number is prime only if it is odd.
If a number is prime then it is odd.
2 EVEN
The Converse
antecedent and the
The converse of a statement switches the __________
consequent
_____________
The converse of “If it is a tiger then it has stripes” is
If it has stripes, then it is a tiger
“____________________________”.
Zebra
NO! Can you think of a counterexample? __________
Is this true?____.
For each statement below, decide its truth value. If it is false, write or draw a
counter example. Then write the converse and decide its truth value
Statement
If a polygon is
equilateral, the
polygon is regular
An obtuse angle has
a measure between
90° & 180
All ants are insects.
Truth value/
Counterexample
F
T
T
If an animal is an ant,
then it is an insect.
If a figure is a
pentagon, then it is a
decagon
F
Converse
Truth Value/
Counterexample
If a polygon is
regular, then it
is equilateral.
T
If an angle is between
90 & 180, then it is an
obtuse angle
T
If an animal is an
insect, then it is an
ant.
F
If a figure is a
decagon, then it is a
pentagon
F
beetle
Is the truth value of the converse always the same as the truth value of the
original statement?
No!
Biconditional Statements
When both a statement and its converse are true, we can put them
together into one statement called a bicondtional statement.
A biconditional statement uses the phrase “if and only if” between its
two parts. Here is an example.
Two lines are perpendicular if and only if they
intersect to form a right angle.
This means BOTH,
• If two line are perpendicular, then they intersect to form a right
angle. AND the converse
• If two lines intersect to form a right angle, then they are
perpendicular
Note: the symbol for “perpendicular” in the language of
geometry is 
Biconditional Statements cont.
In the converses that you wrote in the boxes on page 2, there was
only one example where both the statement and the converse were
true.
Notice that it was the definition of an obtuse angle.
A good definition can always be written as a
biconditional statement.
What biconditional statement can be made from this statement and
its converse?
If two angles are supplementary, then their measures sum to 180°
Converse:
If two angles have measures that sum to 180°, then they are
supplementary
Biconditonal:
Two angles are supplementary if and only if their measures sum to
180°
Biconditional Statements cont.
What two statements can be made from these biconditional
statements ?
BC and BA are opposite rays if and only if B is between A and C
If-Then
If BC and BA are opposite rays, then B is between A & C
Converse
If B is between A & C, then BC & BA are opposite rays
Biconditional Statements cont.
What two statements can be made from these biconditional
statements ?
A polygon is a triangle if and only if it has three sides.
If- Then
If a polygon is a triangle, then it has three sides
Converse
If a polygon has three sides, then it is a triangle.