Download Conditional statement

2.3 CONDITIONAL
STATEMENTS
Geometry R/H
Conditional Statement
• A Conditional statement is a statement that can be written in
the form:
If P, then Q.
The hypothesis is the P part of a conditional statement
following the word if.
• The conclusion is the Q part of a conditional statement
following the word then.
Venn Diagram

A Venn diagram is a diagram of interlocking circles used to help solve
problems.
• Here is a Venn diagram that
represents the conditional
statement
If P then Q.
Q
P
What is the hypothesis and conclusion for
each of these conjectures?
• If today is July 4, then it is Independence Day in the
United States.
• If it is sunny outside, then I will go to the park.
Sometimes the hypothesis and conclusion are switched.
• I will go to the park if it is sunny outside.
Steps in Writing a Conditional Statement
Write a conditional statement from the following:
An obtuse triangle has exactly one obtuse angle.
1.
Identify the hypothesis and the conclusion.
An obtuse triangle
has exactly one obtuse angle.
2. Make a complete sentence from each part.
If a triangle is obtuse, then it has exactly one obtuse
angle.
Writing a Conditional Statement
• Write a conditional statement from the following Venn
diagram.
Birds
Blue Jays
If an animal is a
blue jay, then it is a
bird.
The inner oval represents the hypothesis, and the outer oval
represents the conclusion.
Truth Value
• A conditional statement has a truth value of either true or
false.
• It is false only when the hypothesis is true and the
conclusion is false.
• Consider the statement “If it rains, we will leave early”.
• If it doesn’t rain, does that prove the statement is false?
Analyzing the Truth Value
• For the conditional statement “If it rains, we will leave
early”, what would make it false?
• To show that a conditional statement is false, you need to
find only one counter-example where the hypothesis is
true and the conclusion is false.
Negation of a Statement
• The negation of a statement is the statement formed by adding
not to the original statement
• The negation of p is “not p”, written as ∼p.
• The negation of a statement such as “It is sunny today” is “It
is not sunny today”.
• The negation of a true statement is false, the negation of a
false statement is true.
Related Conditionals
Definitions
• A conditional statement is written in the
form
Symbols
p →q
“if p then q”.
• The converse is the statement formed
by exchanging the hypothesis and
conclusion.
q →p
Give the converse for these:
Conditional Statement: If the opposite sides of a
quadrilateral are congruent, then the quadrilateral is
a parallelogram.
Converse: If a quadrilateral is a parallelogram, then
the opposite sides of the quadrilateral are
congruent.
Conditional Statement: If two angles are complementary, then they are acute angles.
Converse: If two angles are acute, then they are
complementary.
Related Conditionals
Definitions
• The inverse is the statement formed by
negating the hypothesis and conclusion.
• The contrapositive is the statement
formed by both exchanging and negating
the hypothesis and conclusion.
Symbols
∼p → ∼q
∼q → ∼p
Give the inverse for these:
Conditional Statement: If a figure is a square, then it has four
right angles.
Inverse: If a figure is not a square, then it does not have four
right angles.
Conditional Statement: If an angle is a right angle, then it is
not an acute angle.
Inverse: If an angle is not a right angle, then it is an acute
angle.
Examples of Contrapositive:
Conditional Statement: If I see lightning, then it is
raining.
Contrapositive: If it is not raining, then I do not
see lightning.
Conditional Statement: If a figure is a rhombus,
then the diagonals are perpendicular.
Contrapositive: If a figure does not have
perpendicular diagonals, then it is not a
rhombus.
Example – Conditionals
• Write the converse, inverse, and contrapositive of
the statement “If a figure is a square, then it is a
rectangle.”
• Converse: “If a figure is a rectangle, then it is a
square.”
• Inverse: “If a figure is not a square, then it is not a
rectangle.”
• Contrapositive: “If a figure is not a rectangle, then it
is not a square”.
Example – Truth Value
• Find the truth value of the following statements.
• Conditional Statement: “If a figure is a square, then
true
it is a rectangle.”
• Converse: “If a figure is a rectangle, then it is a
square.” false
• Inverse: “If a figure is not a square, then it is not a
rectangle.”
false
• Contrapositive: “If is not a rectangle, then it is not a
square.
true
Excerpt from Alice
in Wonderland
“Then you should say what you mean,” the March
Hare went on.
“I do,” Alice hastily replied; “at least—at least I mean
what I say—that’s the same thing you know.”
“Not the same thing a bit!” said that Hatter. “Why,
you might just say that ‘I see what I eat’ is the
same thing as ‘I eat what I see!”
Are the truth value’s for the Hatter’s statement and
converse the same?
Logically Equivalent
• In our example, the conditional statement and the
contrapositive were both true, and the converse
and the inverse were both false.
• Two related conditional statements that have the
same truth value are called logically equivalent
statements.
• The conditional and contrapositive (both true), the
inverse and converse (both false) are pairs of
logically equivalent statements.
If there is snow on the ground, then
flowers are not in bloom.
Write the
• a) inverse
• b) converse
• c) contrapositive
Biconditional
• Biconditional- a combination of a conditional and its
converse using the word ‘and,’ if they are both true.
• If a polygon has 4 sides, then it is a quadrilateral, and, if
a it is a quadrilateral, then a polygon has 4 sides.
• The phrase ‘if and only if’ can be used to shorten
biconditionals.
• A polygon has 4 sides if and only if it is a quadrilateral.
Summary of Conditionals