Mathematics In the Modern World

1. Problem Solving
with Logic
by: GROUP 3

2. At the end of the presentation, you should be able to:
Objectives:
• Use different types of reasoning to justify statements
and arguments.
• Solve problems involving patterns and problems
following Polya’s Strategy.
• Organize methods and approaches for solving
problems.

3. ELEMENTARY LOGIC
Logic is technically defined as "the science or study
of how to evaluate arguments and reasoning. Logic
helps us to differentiate correct reasoning from
poor reasoning. It is important in the sense that it
helps us to reason correctly. Others defined logic
as discipline that deals with the methods of
reasoning.

4. Inductive Reasoning
- the process of reaching a general conclusion by
examining specific samples. The conclusion
formed by using inductive reasoning is often
called a conjecture, since it may or may not be
correct.

5. Example
a. 3, 6, 9, 12, 15, ?
b. 1, 3, 6, 10, 15, ?

6. Inductive reasoning is not used just to
predict the next number in a list. We
can also use inductive reasoning to
make a conjecture about an arithmetic
procedure.

7. Example:
Pick a number. Multiply the
number by 8, add 6 to the
product, divide the
sum by 2, and subtract 3.

8. Solution:
Original No.: 5
Multiply by 8: 8 x 5 = 40
Add 6: 40 + 6 = 46
Divided by 2: 46 ÷ 2 = 23
Subtract 3: 23 - 3 = 20

9. Evaluation
Inductive Reasoning
to Solve an
Application
a. If a pendulum has a
length of 25 units,
what is its period?
b. If the length of a
pendulum is quadrupled,
what happens
to its period?

10. Conclusions based on inductive reasoning may be incorrect.
As an illustration, consider the circles shown below.

13. Counterexamples
A statement is a true statement if and only if it is
true in all cases. If you can find one case for
which a statement is not true, called a
counterexample, then the statement is a false
statement.

14. Example: Verify that
each of the
following
statements is a false
statement by
finding a
counterexample.
a. |x|> 0
b. x^2 > x
c. √x^2 = x

15. a. |x|> 0
a. Let x=0. Then |x|>0.
Because 0 is not
greater than 0, we
have found a
counterexample. Thus
“for all x, |x|>0 ” is
a false statement.

16. b. x^2 > x
b. For x=1, we have
1^2=1. Since 1 is not
greater than 1, we
have found a
counterexample. Thus
“for all x,x2>x” is
a false statement.

17. c. √x^2 = x
c. Consider x=−3. Then
√(−3)^2=√9=3. Since 3 is
not
equal to −3 , we have
found a counterexample.
Thus “for
all x,√x2=x” is a false
statement.

18. Deductive Reasoning
- distinguished from inductive reasoning in that it
is the process of reaching a conclusion by applying
general principles and procedures. Also the
process of reaching a conclusion by applying
general assumptions, procedures, or principles.

19. Example:
Pick a number. Multiply the
number by 8, add 6 to the
product, divide the
sum by 2, and subtract 3.

20. Solution:
Multiply by 8: 8n
Add 6 to the product: 8n + 6
Divided the sum by 2: 8n + 6 /
2 = 4n + 3
Subtract 3: 4n + 3 - 3 = 4n

21. Determine whether
each of the
following arguments
is an example of
inductive reasoning
or deductive
reasoning.
a. During the past 10 years, a tree
has produced plums every other
year. Last year the tree did not
produce plums, so this year the tree
will produce plums.
b. All home improvements cost more
than the estimate. The contractor
estimated my home improvement will
cost $35,000. Thus my home
improvement will cost more than
$35,000.

22. Logical Puzzles
Some logic puzzles can be solved by using
deductive reasoning and a chart that
enables us to display the given information
in a visual manner.

23. Example:
Each of four neighbors,
Sean, Maria, Sarah, and
Brian,
has a different occupation
(editor, banker, chef, or
dentist).
From the following clues,
determine the occupation
of each neighbor.
(1) Maria gets
home from
work after the
banker but
before the
dentist.
(2) Sarah, who
is the last to
get home from
work, is not the
editor.
(3) The dentist
and Sarah
leave for work
at the same
time.
(4) The
banker lives
next door
to Brian.

24. From clue 1,
Maria is not
the banker or
the dentist.
Write X1.
From clue 2,
Sarah is not
the editor.
Write X2.
From clue 3,
Sarah is not
the dentist.
Write X3.
From clue 4,
Brian is not
the banker.
Write X4.

25. Solution: From clue 4, Brian is not the banker. Write X4 for this
condition.

26. Problem Solving with Patterns
TERMS OF A SEQUENCE
A sequence is an ordered list of numbers. The
numbers in a sequence that are separated by
commas are the terms of the sequence.

27. Types of
Sequence
Arithmetic
Quadratic
Cubic

28. Sequence
Classification
(1) Finite Sequence
Ex. Set of prime numbers
below 20
(2) Infinite Sequence
Ex. A set of natural numbers

29. Order
of
Sequence
(1) Ascending Order
Ex. 1, 2, 3, 4, 5
(2) Descending
Order
Ex. 5, 4, 3, 2, 1

30. Example:
2, 5, 8, 11, 14, . .

31. Difference table
- shows the differences between successive terms
of the sequence.
Example:
2, 5, 8, 11, 14, . ..

32. Example:
5, 14, 27, 44, 65, . . .

33. Example:
2, 7, 24, 59, 118, 207, . . .

34. nth Term Formula for a
Sequence
In some cases we can use patterns to predict a
formula, called an nth term formula, that generates
the terms of a sequence.
We will often use the letter n to represent an
arbitrary natural number.

35. Example #1:
2, 5, 8, 11,14, ...

36. Example #1:
2, 5, 8, 11,14, ...

37. Example #2:
5, 14, 27, 44, 65, ...

38. Example #2:
5, 14, 27, 44, 65, ...

39. Example #2:
5, 14, 27, 44, 65, ...

40. Example #3:
2, 7, 24, 59, 118, 207...

41. Example #3:
2, 7, 24, 59, 118, 207...

42. Example #3:
2, 7, 24, 59, 118, 207...

43. Polya’s Problem-Solving Strategy
One of the foremost recent mathematicians to
make a study of problem solving was George Polya
(1887–1985). He was born in Hungary andmoved to
the United States in 1940. The basic problem-
solving strategy that Polya advocated consisted of
the following four steps:

44. Polya’s Four-Step Problem-Solving
Strategy
1.
Understand
the
Problem
2.
Devise
a Plan
3.
Carry
Out the
Plan
4.
Review
the
Solution

45. Example:
Consider the map shown
beside.
Allison wishes to
walk along the streets
from point A to point B.
How many
direct routes can Allison
take?

46. Solution:
Understand the Problem. We
would not be able to answer
the question if Allison retraced
her path or traveled away from
point B. Thus we assume that
on a direct route, she always
travels along a street in a
direction that gets her closer
to point B.

47. Solution:
Devise a Plan. The
map given above has
many extraneous
details. Thus we
make a diagram that
allows us to
concentrate on the
essential information.

48. Solution:
Carry Out the Plan. Using
the pattern discovered
on the previous page, we
see from the figure at
the left that the number
of routes from point A to
point B is 20 +15 =35.

49. Solution:
Review the
Solution. Ask
yourself whether a
result of 35
seems reasonable.

50. Work Principle

51. Steps in
Problem Solving
• Familiarize
2. Translate
3. Solve
4. Check
5. State

52. Example

54. Example

56. Thank You!
SEE YOU NEXT TIME!