- 1. Obj. 9 Inductive Reasoning Objectives: The student is able
to (I can): Use inductive reasoning to identify patterns and make
conjectures Find counterexamples to disprove conjectures Identify,
write, and analyze the truth value of conditional statements. Write
the inverse, converse, and contrapositive of a conditional
statement.
2. Find the next item in the sequence: 1. December, November,
October, ... SeptemberSeptemberSeptemberSeptember 2. 3, 6, 9, 12,
... 15151515 3. , , , ... 4. 1, 1, 2, 3, 5, 8, ... 13131313 This is
called the FibonacciThis is called the FibonacciThis is called the
FibonacciThis is called the Fibonacci
sequence.sequence.sequence.sequence. 3. inductive reasoning
conjecture Reasoning that a rule or statement is true because
specific cases are true. A statement believed true based on
inductive reasoning. Complete the conjecture: The product of an odd
and an even number is ______ . To do this, we consider some
examples: (2)(3) = 6 (4)(7) = 28 (2)(5) = 10 eveneveneveneven 4.
counterexample If a conjecture is true, it must be true for every
case. Just one exampleJust one exampleJust one exampleJust one
example for which the conjecture is false will disprove it. A case
that proves a conjecture false. Example: Find a counterexample to
the conjecture that all students who take Geometry are 10th
graders. 5. Examples To Use Inductive Reasoning 1. Look for a
pattern. 2. Make a conjecture. 3. Prove the conjecture or find a
counterexample to disprove it. Show that each conjecture is false
by giving a counterexample. 1. The product of any two numbers is
greater than the numbers themselves. ((((----1)(5) =1)(5) =1)(5)
=1)(5) = ----5555 2. Two complementary angles are not congruent. 45
and 4545 and 4545 and 4545 and 45 6. Sometimes we can use inductive
reasoning to solve a problem that does not appear to have a
pattern. Example: Find the sum of the first 20 odd numbers. Sum of
first 20 odd numbers? 1 1 + 3 1 + 3 + 5 1 + 3 + 5 + 7 1 4 9 16 12
22 32 42 202 = 400 7. These patterns can be expanded to find the
nth term using algebra. When you complete these sequences by
applying a rule, it is called a functionfunctionfunctionfunction.
Examples: Find the missing terms and the rule. To find the pattern,
the coefficient of n is the difference between each term, and the
value at 0 is what is added or subtracted. 1 2 3 4 5 8 20 n -3 -2
-1 0 1 4 16 n 4 1 2 3 4 5 8 20 n 32 39 46 53 60 81 165 7n+25 8.
conditional statement hypothesis conclusion A statement that can be
written as an if-then statement. Example: IfIfIfIf today is
Saturday, thenthenthenthen we dont have to go to school. The part
of the conditional following the word if. today is Saturday is the
hypothesis. The part of the conditional following the word then. we
dont have to go to school is the conclusion. 9. Notation Examples
Conditional statement: p q, where p is the hypothesis and q is the
conclusion. Identify the hypothesis and conclusion: 1. If I want to
buy a book, then I need some money. 2. If today is Thursday, then
tomorrow is Friday. 3. Call your parents if you are running late.
10. Examples To write a statement as a conditional, identify the
sentences hypothesis and conclusion by figuring out which part of
the statement depends on the other. Write a conditional statement:
Two angles that are complementary are acute. If two angles are
complementary, then theyIf two angles are complementary, then
theyIf two angles are complementary, then theyIf two angles are
complementary, then they are acute.are acute.are acute.are acute.
Even numbers are divisible by 2. If a number is even, then it is
divisible by 2.If a number is even, then it is divisible by 2.If a
number is even, then it is divisible by 2.If a number is even, then
it is divisible by 2. 11. truth value T if a conditional is true, F
if a conditional is false. The statement is false only when theThe
statement is false only when theThe statement is false only when
theThe statement is false only when the hypothesis is true and the
conclusion ishypothesis is true and the conclusion ishypothesis is
true and the conclusion ishypothesis is true and the conclusion is
false.false.false.false. To show that a conditional is false, you
need only find one counterexample where the hypothesis is true and
the conclusion is false. Hypothesis Conclusion Statement T T T TTTT
FFFF FFFF F T T F F T 12. Examples Determine if each conditional is
true. If false, give a counterexample. 1. If your zip code is
76012, then you live in Texas. TrueTrueTrueTrue 2. If a month has
28 days, then it is February. September also has 28 days,
whichSeptember also has 28 days, whichSeptember also has 28 days,
whichSeptember also has 28 days, which proves the conditional
false.proves the conditional false.proves the conditional
false.proves the conditional false. 3. If 14 is evenly divisible by
3, then tomorrow is Tuesday. The hypothesis is false, so theThe
hypothesis is false, so theThe hypothesis is false, so theThe
hypothesis is false, so the conditional isconditional isconditional
isconditional is truetruetruetrue.... Texas 76012 13. negation of p
Not p Notation: ~p Example: The negation of the statement Blue is
my favorite color, is Blue is notnotnotnot my favorite color.
Related Conditionals Symbols Conditional p q Converse q p Inverse
~p ~q Contrapositive ~q ~p 14. Example Write the conditional,
converse, inverse, and contrapositive of the statement: A cat is an
animal with four paws. Type Statement Truth Value Conditional (p q)
If an animal is a cat, then it has four paws. T Converse (q p) If
an animal has four paws, then it is a cat. F Inverse (~p ~q) If an
animal is not a cat, then it does not have four paws. F Contrapos-
itive (~q ~p) If an animal does not have four paws, then it is not
a cat. T 15. Example Write the conditional, converse, inverse, and
contrapositive of the statement: When n2 = 144, n = 12. Type
Statement Truth Value Conditional (p q) If n2 = 144, then n = 12. F
(n = 12) Converse (q p) If n = 12, then n2 = 144. T Inverse (~p ~q)
If n2 144, then n 12 T Contrapos- itive (~q ~p) If n 12, then n2
144 F (n = 12)