Transcript 2.2 Analyze Conditional Statements

Analyze Conditional Statements
Objectives:
1. To write a conditional statement in if-then
form
2. To write the negation, converse, inverse,
and contrapositive of a conditional
statement and identify its truth value
3. To write a biconditional statement
Example 1
What are Clairzaps?
Conditionals
Conditionals are statements written in ifthen form.
Subject
Predicate
A hexagon is a polygon with six sides.
If it is a hexagon, then it is a polygon with six sides.
-OR- For clarity:
If a polygon is a hexagon, then it has six sides.
Hypothesis
Conclusion
Example 2
Rewrite the conditional statement in if-then
form.
All 90° angles are right angles.
Example 3
Rewrite the conditional statement in if-then
form.
Two angles are supplementary if they
are a linear pair.
Converse
The converse of a conditional is formed by
reversing the hypothesis (if) and
conclusion (then).
Example 4
Write the following statement in if-then form,
then write its converse. Is the converse
always true?
All squares are rectangles.
Truth Value
A conditional statement can be true or false.
• True: To show that a conditional is true,
you have to prove that the conclusion is
true every time the hypothesis is satisfied.
• False: To show a conditional is false, you
just have to find one example in which the
conclusion is not true when the hypothesis
is satisfied.
Example 5
What is the opposite of the following
statements?
1. The ball is red.
2. The cat is not black.
Negation
The negation of a statement is the opposite
of the original statement.
Statement: The sick boy eats meat.
Negation: The sick boy does not eat meat.
Notice that only the verb of the sentence
gets negated.
Symbolic Notation
Mathematicians are notoriously lazy, creating
shorthand symbols for everything. Conditional
statements are no different.
Symbol
Concept
p
Original Hypothesis
q
Original Conclusion
→
“Implies”
~
“Not”
p→q
“p implies q” “if p, then q”
~p
“not p”
All Kinds of Conditionals
So the symbols make conditionals easy and fun!
Statement
Symbols
Conditional
p→q
Converse
q→p
Inverse
~p → ~q
Contrapositive
~q → ~p
All Kinds of Statements
Here are some examples of writing the
converse, inverse, and contrapositive of a
conditional statement.
Example 6
Write the converse, inverse, and
contrapositive of the conditional statement.
Indicate the truth value of each statement.
If a polygon is regular, then it is equilateral.
Which of the statements that you wrote are
equivalent?
Equivalent Statements
When pairs of statements are both true or
both false, they are called equivalent
statements.
• A conditional and its contrapositive are
equivalent.
• An inverse and the converse are
equivalent.
– So if a conditional is true, so its contrapositive.
Definitions in Geometry
In geometry, definitions can be written in ifthen form. It is important that these
definitions are reversible. In other words,
the converse of a definition must also be
true.
If a polygon is a hexagon, then it has exactly six sides.
-ANDIf a polygon has exactly six sides, then it is a hexagon.
Perpendicular Lines
If two lines intersect
to form a right
angle, then they
are perpendicular
lines.
Example 7
Write the converse
of the definition
of perpendicular
lines.
If two lines intersect
to form a right
angle, then they
are perpendicular
lines.
Biconditional
A biconditional is a statement that
combines a conditional and its true
converse in “if and only if” form.
If a polygon is a hexagon, then it has exactly six sides.
-ANDIf a polygon has exactly six sides, then it is a hexagon.
A polygon is a hexagon if and only if
it has exactly six sides.
Example 8
Write the definition
of perpendicular
lines as a
biconditional
statement.
If two lines intersect
to form a right
angle, then they
are perpendicular
lines.
Exercise 9
Rewrite the definition of right angle as a
biconditional statement.