Transcript File

Geometry - Lesson 2.2
TODAY’S OBJECTIVE:
Standard: MM1G2
Students will understand and use the language of
mathematical argument and justification.
a.
b.
Use conjecture, inductive reasoning, deductive
reasoning, counterexamples, and indirect proof as
appropriate.
Understand and use the relationships among a
statement and its converse, inverse, and
contrapositive.
Essential Question
OWhat is logic?
Conditionals (If-then form)
O Conditionals are if-then statements
O Ex: If you are fourteen, then you are a
teenager.
O Ex: If x=10, then 2x=20.
Converse, Inverse, & Contrapositive
O Statement: if p then q
O Converse: if q then p
O Statement: If you do your
math homework, then you
get a good grade.
O Inverse: if not p then not q
O Converse: If you get a
good grade, then you do your
math homework.
O Contrapositive: if not q
O Inverse: If you don’t do
your math homework, then
you don’t get a good grade.
then not p
O Contrapositive: If you
don’t get a good grade, then
you don’t do your math
homework.
Negation
O The negation of statement p is "not p."
O The negation of p is symbolized by "~p."
Now, you try. Negate the following statement:
If you can eat something, then it is considered food.
Negation:
If you cannot eat something, then it is not considered food.
Biconditional
O A biconditional statement is defined to be true
whenever both parts have the same truth value.
O The biconditional operator is denoted by a
double-headed arrow
O The biconditional p
.
q represents:
"p if and only if q," where p is a hypothesis and q
is a conclusion.
Inductive & Deductive Reasoning
O Also known as logical reasoning
O Systems for reaching logical conclusions
Inductive reasoning: the process of arriving at a conclusion
based on a set of observations.
In itself, it is not a valid method of proof.
Deductive reasoning: the process of arriving at a conclusion
based on previously known facts.
It is the way proofs are written.
Law of Syllogism
O The Law of Syllogism:
O If
and
then
p → q is true,
q → r is true,
p
r is also true.
O Use the Law of Syllogism to come to a
conclusion in the following example.
Example:
O Given 1: If I study and work hard, then I get
good grades.
O Given 2: If I get good grades, then I get into a
good college.
O Therefore: If I study and work hard...
... I get into a good college!
Practice Time!
O In the textbook,
O Do page 207 Set A Odd Numbers.
O Do page 209 Set B Even Numbers.
O Raise your hand, and I will come
around if you need help!
Exit Slip!
O 1. Give an example of a conditional statement.
O 2. State the contrapositive of the following
statement: If you go shopping, then you have
money.
O 3. If I know that leather is made from cows, and my
chair is made of leather, then my chair was made
from a cow.
O What kind of reasoning did I use here?