6-2 Reasoning and Proof6-2 Reasoning and Proof

"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 the Hatter, "Why, you might just as well say that 'I see what I eat' is

the same thing as 'I eat what I see'!"

"You might just as well say," added the March Hare, "that 'I like what I get' is the same thing as 'I get what I like'!“

"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'!“

"It is the same thing with you," said the Hatter, and here the conversation dropped, and the party sat silent for a minute.

6-2 Conditional Statements6-2 Conditional Statements

1.1. A _________________ is a statement A _________________ is a statement that can be expressed in that can be expressed in ________________form.form.

conditional statementconditional statement““if-then”if-then”

2.2. A conditional statement has A conditional statement has __________________..The The ____________________ is the is the ________ part. part.The The ____________________ is the is the ____________ part. part.

hypothesishypothesistwo partstwo parts

““if”if”conclusionconclusion ““then”then”

6-2 Conditional Statements6-2 Conditional Statements

Example: Example:

(Original) I breathe when I sleep

(Conditional) If I am sleeping, then I am breathing.

6-2 Conditional Statements6-2 Conditional Statements

To fully analyze this conditional statement, To fully analyze this conditional statement, we need to find three new conditionals:we need to find three new conditionals:

ConverseConverse

InverseInverse

ContrapositiveContrapositive

6-2 Conditional Statements6-2 Conditional Statements

The The ________________ of a conditional statement of a conditional statement is formed by is formed by switchingswitching the hypothesis and the hypothesis and the conclusion.the conclusion.

Example: Example:

converseconverse

(Conditional)(Conditional) If If I am sleepingI am sleeping, then , then I amI am breathingbreathing..

(Converse)(Converse) If I am breathing, then I am If I am breathing, then I am sleeping. sleeping.

6-2 Conditional Statements6-2 Conditional Statements

The The ________________ of a conditional statement of a conditional statement is formed by negating (inserting “not”) the is formed by negating (inserting “not”) the hypothesis and the conclusion.hypothesis and the conclusion.

Example: Example:

inverseinverse

(Conditional)(Conditional) If If I am sleepingI am sleeping, then , then I amI am breathingbreathing..

(Converse)(Converse) If I am If I am notnot sleeping, then sleeping, then I am I am notnot breathing. breathing.

6-2 Conditional Statements6-2 Conditional Statements

The The ____________________________ of a conditional statement of a conditional statement is formed by negating the hypothesis and the is formed by negating the hypothesis and the conclusion conclusion of the converse.of the converse.

Example: Example:

(Converse)(Converse) If I am breathing, then I am If I am breathing, then I am sleeping. sleeping.

(Contrapositive)(Contrapositive) If I am If I am notnot breathing, then I breathing, then I am am notnot sleeping. sleeping.

contrapositive

6-2 Conditional Statements6-2 Conditional Statements

ConditionalConditional

( )( )

InverseInverse

( )( )

ConverseConverse

( )( )

ContrapositiveContrapositive

((

))

If I am sleeping, then I am If I am sleeping, then I am breathingbreathing..

If I am If I am not not sleeping, then I am sleeping, then I am not not breathing.breathing.

If I am breathing, then I am If I am breathing, then I am sleeping.sleeping.

If I am If I am notnot breathing, then I breathing, then I am am notnot sleeping. sleeping.

if…then if…then

insert notinsert not

switchswitch

switch and switch and insert not insert not

6-2 Conditional Statements6-2 Conditional Statements

The conditional statement, inverse, The conditional statement, inverse, converse and contrapositive all have a converse and contrapositive all have a truth value. That is, we can determine if truth value. That is, we can determine if they are they are truetrue or or falsefalse..

When two statements are both true When two statements are both true oror both false, we say that they are both false, we say that they are logically logically equivalentequivalent. .

6-2 Conditional Statements6-2 Conditional StatementsConditionalConditional If If mm<A = 30<A = 30°°, then <A is , then <A is

acute.acute.

InverseInverse

(insert not)(insert not)

ConverseConverse

(switch)(switch)

ContrapositiveContrapositive

(switch then (switch then insert not)insert not)

TT

TT

FF

FF

If If mm<A <A ≠≠ 30 30°,°, then <A is then <A is not not acute.acute.

If <A is acute, then If <A is acute, then

mm<A = 30<A = 30°°..

If <A is If <A is not not acute, then acute, then mm<A <A ≠≠ 30 30°°..

6-2 Conditional Statements6-2 Conditional Statements

The conditional statement and its The conditional statement and its contrapositive have the same truth value. contrapositive have the same truth value.

They are They are both trueboth true. .

They are logically equivalent.They are logically equivalent.

6-2 Conditional Statements6-2 Conditional Statements

The inverse and the converse have the The inverse and the converse have the same truth value. same truth value.

They are They are both falseboth false. .

They are logically equivalent.They are logically equivalent.

PracticePractice

Translate the following statement into a Translate the following statement into a conditional statement. Then find the conditional statement. Then find the converse, inverse and contrapositive.converse, inverse and contrapositive.

““A cloud of steam can be seen when the A cloud of steam can be seen when the space shuttle is launchedspace shuttle is launched””

1. Identify the underlined portion 1. Identify the underlined portion of the conditional statement.of the conditional statement.

A.A. hypothesishypothesis

B.B. ConclusionConclusion

C.C. neitherneither

3. Identify the underlined portion 3. Identify the underlined portion of the conditional statement.of the conditional statement.

A.A. hypothesishypothesis

B.B. ConclusionConclusion

C.C. neitherneither

4. Identify the converse for the 4. Identify the converse for the given conditional.given conditional.

A.A. If you do not like tennis, then you do not If you do not like tennis, then you do not play on the tennis team.play on the tennis team.

B.B. If you play on the tennis team, then you like If you play on the tennis team, then you like tennis.tennis.

C.C. If you do not play on the tennis team, then If you do not play on the tennis team, then you do not like tennis.you do not like tennis.

D.D. You play tennis only if you like tennis.You play tennis only if you like tennis.

5. Identify the inverse for the 5. Identify the inverse for the given conditional.given conditional.

A.A. If 2x is not even, then x is not odd.If 2x is not even, then x is not odd.

B.B. If 2x is even, then x is odd.If 2x is even, then x is odd.

C.C. If x is even, then 2x is odd.If x is even, then 2x is odd.

D.D. If x is not odd, then 2x is not even.If x is not odd, then 2x is not even.