Notes 2.3 GeoN.notebook
September 17, 2013
Notes Section 2.3 ‐ Analyze statements in if­then form. Write the converse, inverse and contrapositive of if­then statements.
conditional statement ‐ a sentence written in "if‐then" form
hypothesis ‐ the part of the conditional statement that follows the "if"
conclusion ‐ the part of the conditional statement that follows the "then"
converse statement ‐ switches the hypothesis and conclusion
inverse statement ‐ negates the conditional statement
contrapositive statement ‐ negates the converse
biconditional statement ‐ when a conditional statement and its converse are both true. Conditional statement is written without the if and then and "if and only if" or "iff" separates the hypothesis and conclusion.
logically equivalent ‐ two conditional statements that have the same truth value
Identify the hypothesis and conclusion of the following statement.
1. If today is Monday, then tomorrow is Tuesday.
hypothesis:
conclusion:
2. You will cry if you are a baby.
Write the statement in the if­then form.
3. A square has four sides.
4. Dogs have 4 legs.
Which of the following is the correct if­then form of the given statement?
5. A polygon with 8 sides is an octagon.
A. If an octagon has 8 sides, then it is a polygon.
B. If a polygon has 8 sides, then it is an octagon.
C. If a polygon is an octagon, then it has 8 sides.
D. none of the above
Notes 2.3 GeoN.notebook
September 17, 2013
The hypothesis and conclusion of a conditional statement can have a truth value of true or false, as can the conditional statement itself.
example: If Tom inishes his homework , then he will clean his room.
Hypothesis Tom inishes his homework
T
Conclusion
he will clean his room
T
Conditional
If Tom inishes his homework , then he will clean his room.
T
Tom does his homework and cleans his room
T
F
F
Tom does his homework but doesn't clean his room
F
F
T
F
?
?
The conditional only indicates what will happen
if Tom does inish his homework. He could clean his room or not clean his room if he does not inish
his homework.
When the hypothesis of a conditional is not met, the truth of a conditional cannot be determined. When the truth of a conditional statement cannot be determined it is considered true by default. To show that a conditional statement if true, show that every time the hypothesis is true, the conclusion is true. To show a conditional statement is false, you only need to ind one counterexample for which the hypothesis is true but the conclusion is false.
Determine the truth value of the conditional statement. If false, give a counterexample.
6.
If you subtract a whole number from another whole number, the result is also a whole number.
7.
If an animal is a dog, then it has four legs.
8.
The product of whole numbers is greater than or equal to 0.
9.
When a rectangle has an obtuse angle, it is a parallelogram.
10.
If yesterday was Tuesday, then today is Monday.
11.
If a triangle has four right angles, then it is a rectangle.
Notes 2.3 GeoN.notebook
September 17, 2013
Related Conditionals
conditional statement ‐ If a then b
Write the converse, inverse, and converse ‐ If b then a
contrapositive statements. Determine the inverse ‐ if not a then not b
contrapositive ‐ if not b then not a
truth value of each statement. If a biconditional ‐ a iff b
statement is false, give a counterexample.
12. Bats are animals that can fly.
If‐then form: Converse:
Inverse:
Contrapositive:
Example 4
13. Write the converse, inverse, and contrapositive of the following statement. Determine the truth value of each statement. If a statement is false, give a counterexample.
If 2 <’s have the same measure then they are ≅.
Converse:
Inverse:
Contrapositive:
Notes 2.3 GeoN.notebook
September 17, 2013
Biconditional Statement – the conditional statement and the converse are both true. written: a iff b read: a if and only if b
Conditional: If today is Monday then tomorrow is Tuesday.
Converse: If tomorrow is Tuesday, then today is Monday.
Biconditional: 14.
Write the converse of "If two angles are complementary then the sum of their measures is 90." Determine truth values of both statements. If both true then write the biconditional.
Converse:
Biconditional: