Math 31 Reviewer and Pointers

Eric G. Lauron
September 18, 2021

Disclaimer: This material is intended only for our review to enhance our preparedness for
the incoming exam. Please circulate this around our section only. Some items are not answered.
Please be diligent in answering them.

1 Topic 1: Mathematics in Nature

Definition 1 A logical statement is a declarative sentence which conveys factual information.
It is either true or false but not both. If the information is correct then we say that the statement
is true; and if the information is incorrect, then we say that the statement is false.

Explain why the sentence below is a logical statement or not.

1. All prime numbers are odd.
This is a logical statement since it is a declarative sentence that conveys a factual information.
The truth value is false.

2. x + y
This is not a declarative sentence.

3. Open the door.

This is a command.

4. The mayor is doing a great job.

5. What is it?

Definition 2 The truth value of a statement is true (T) if the statement is true and false (F)
if it is false.

Definition 3 A logical statement can either be simple or compound. A simple statement is a

logical statement carrying one piece of information while a compound statement is a statement
that is connected with one or more words or phrases such as and, or, not, if...then, if and only
if called connectives.

Theorem 1 The following are the truth values for the compound statements:
1. The truth value of a negation is the opposite truth value of the original statement.

2. The conjunction p ∧ q is only true if both p and q are true.

1
 3. The disjunction p ∨ q is false only if both p and q are false.

4. The conditional p → q is false only if p is true and q is false.

5. The biconditional p ⇐⇒ q is true only if both p and q have the same value.

Suppose p and q are logical statements. The following table is a comprehensive presentation
for the truth value of the compound statements.

p q ¬p p ∧ q p ∨ q p → q p ⇐⇒ q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T
Find the truth value of the following:
1. Cebu city is in region V III and region V III is in the Philippines.
Solution:
Let
p : Cebu city is in region V III;
q : region V III is in the Philippines.
We have p is false and q is true. Therefore, by conjunction, the statement is false.

2. For any x ∈ R, it is either x is negative or x is nonnegative.

Solution:
Let p be the statement, x is negative and q be the statement x is nonnegative. For any
x ∈ R:
Case 1: x < 0
p is true or q is false. Therefore, by disjunction, the statement is true.
Case 2: x ≥ 0
p is false or q is true. By disjunction, the statement is true.
Therefore, in both cases, the statement is true.

3. All squares are rhombi if and only if all squares are rectangles.

Definition 4 A truth table is a table that shows the truth value of a statement for all possible
truth values of its components.

Question:
If there are n number of statements, how may possible combination of truth values?
Answer:
Suppose there are n statements. Each statement has two possible values. Hence, we have

2| · 2 · 2 ·{z2 · · · 2 · 2}

of length n. There are 2n possible combination of truth values.

How to Construct a Truth Table

2
 1. Construct a table with 2n +1 rows and n + number of distinct compound statements columns
where n is the number of given statements. Reserve the top row for the statements and the
rest of the 2n rows for the truth values.

2. Write the symbolic form of the statements piece by piece. The first n columns should be the
given statements.
n
3. Distribute the possible truth values. To do this, on the first column, write 22 rows of T and
n
F alternately. On the second column, write alternately T and F on 222 number of rows. On
n
third column would be 223 rows of alternate T and F and so on.

4. Do the table according to the connectives.

Construct the truth table for the following:

1. Let p, q and r be logical statements.

[(p → q) ∧ (q → r)] → (p → r).

Solution:
There are 3 simple statements, so the number of rows is 2n + 1 = 23 + 1 = 8 + 1 = 9. There
are 3 simple statements and 5 compound statements namely, (p → q), (q → r), (p → r),
[(p → q) ∧ (q → r)] and [(p → q) ∧ (q → r)] → (p → r). Hence, there are 3 + 5 = 8 columns.
This is how it is constructed.
p q r (p → q) (q → r) (p → r) [(p → q) ∧ (q → r)] [(p → q) ∧ (q → r)] → (p → r)

Next, distribute the values. For p column, we have

p q r (p → q) (q → r) (p → r) [(p → q) ∧ (q → r)] [(p → q) ∧ (q → r)] → (p → r)
T
T
T
T
F
F
F
F
For the q column, we have

3
 p q r (p → q) (q → r) (p → r) [(p → q) ∧ (q → r)] [(p → q) ∧ (q → r)] → (p → r)
T T
T T
T F
T F
F T
F T
F F
F F
For the r column, we have
p q r (p → q) (q → r) (p → r) [(p → q) ∧ (q → r)] [(p → q) ∧ (q → r)] → (p → r)
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F T
Lastly, do the connectives,
p q r (p → q) (q → r) (p → r) [(p → q) ∧ (q → r)] [(p → q) ∧ (q → r)] → (p → r)
T T T T T T T T
T T F T F F F T
T F T F F T F T
T F F F T F F T
F T T T T T T T
F T F T F T F T
F F T T T T T T
F F T T T T T T
2. Verify the truth value of
(p → r) → [(p → q) ∧ (q → r)]
where p, q and r are logical statements.
Definition 5 A tautology is a statement that is always true. A contradiction is a statement
that is always false.
In the case that a statement is true for cases and false on others, then it is called contingency.
Show if the following is a tautology, contradiction or contingency:
1. (p ∧ ¬(q ∧ r)) ∨ (q → r) for logical statements p, q and r.
2. ¬({¬(p ∧ ¬r) ∧ ¬(q ∧ ¬s)} ∧ ¬{¬(p ∨ q) ∨ (r ∨ s)}) for logical statements p, q, r and s.
Definition 6 Two compound statements A and B are logically equivalent if they both have the
same truth value for all possible truth values of their simple statements.
Suppose p and q are statements, show that ¬(p ⇐⇒ q) and [(p ∧ ¬q) ∨ (q ∧ ¬p)] are logically
equivalent.

4
2 Reasoning and Problem Solving
Definition 7 Reasoning is the act of thinking about something in a logical and sensible way.

Definition 8 Inductive Reasoning is a type of reasoning that forms a conclusion based on the
examination of specific examples. The conclusion drawn is called a conjecture.

Definition 9 Deductive Reasoning is the process of using facts, rules, definitions, or properties
to reach a valid conclusion.

Construct your conjecture by using inductive reasoning and prove it by using deductive rea-
soning.

1. What is the total amount if there are n number of 5-peso coins on the base.

2. If there are n number of dots, how many lines should it generate?

3. In a square egg tray, an egg can only be place on the top space between the four eggs. If the
dimension of the tray is 1 by 1, we cannot put an egg on top of it. If the dimension is 2 by
2, we can put another egg on the space between the four eggs. The total number of eggs is
5. If the dimension of the tray is 3 by 3, then we can put 4 eggs on the second layer on the
spaces between the eggs and 1 egg on the third layer on the space between the four eggs on
the second layer. The total number of eggs is now 9+4+1=14. What is the total number of
eggs in an n by n egg tray?

Definition 10 Counterexample is a specific case in which a statement is false. One counterex-

ample is enough to prove that a statement is false.

Give a counter example for the following:

5
 1. All prime numbers are odd.
Solution:
2 is prime, yet 2 is even.
(x+1)(x−1)
2. x+1
= x − 1.
Solution:
For x = −1, the equality does not hold.

3. A real number is either positive or negative.

4. All quadrilaterals have at least one right angle.

5. If x is a number, then x2 is always positive.

6. There is no irrational number between 0 and 1.

2.1 Steps of Polya’s Problem Solving Technique

1. Understand the problem

2. Devise a plan

3. Carry out the plan

4. Review the Solution

Solve the following:

1. Find the 1234567th digit of the decimal given by 0.987654321987654321987654321.... Solution:

Note that the decimal is repeating and there are 9 digits being repeated namely, 987654321.
So we divide 123456789 by 9. Hence, 1234567 9
= 137174 remainder 1. This means that the
th
1234567 × 9 = 123456 digit is 1. The remainder is 1, which is the first digit of the repeating
term 987654321. Therefore, the 1234567th digit is 9.

2. Find the 10010010001st digit of the decimal given by:

0.012345678901234567890123456789....

3. In a survey of 200 people that had just re-turned from a trip to Europe, the following
information was gathered:

(a) 142 visited England

(b) 95 visited Italy
(c) 65 visited Germany
(d) 70 visited both England and Italy
(e) 50 visited both England and Germany
(f) 30 visited both Italy and Germany
(g) 20 visited all three of these countries

6
How many went to none of these countries.
Solution:
The best way to solve this is to use a diagram.

Next, input first the intersection of the three, meaning there 20 who visited all the three
countries.

Next is the intersections of each countries. To get this, subtract them to the intersection of
three. Hence, Those who visited England and Italy ONLY is 70-20=50. Those who visited
England and Germany ONLY is 50-20=30. Those who visited Italy and Germany ONLY is
30-20=10. Plugging these in to the diagram, we have

7
 Finally, we will get the number of those who visited a country only. To get this we have,
we will subtract the figure to the number of those who visited both and all three. Hence,
for England, we have 142-(50+20+30)=42. For Italy, we have 95-(50+10+20)=15. For
Germany, we have 65-(30+20+10)=5. Plugging in, we have

Finally, we take the sum of these and subtract from the 200 people. Hence we have 200-
(42+15+5+50+30+10+21)=200-172=28. Therefore, there are 28 people who went to none
of these three countries.

4. At the end of the day the manager of Blue Baker wanted to know how many pizzas were
sold. The only information he had is listed below. Use the information to determine how
many pizzas were sold.

(a) 3 pizzas had mushrooms, pepperoni, and sausage

(b) 7 pizzas had pepperoni and sausage
(c) 6 pizzas had mushrooms and sausage but not pepperoni
(d) 15 pizzas had two or more of these toppings
(e) 11 pizzas had mushrooms
(f) 8 pizzas had only pepperoni
(g) 24 pizzas had sausage or pepperoni
(h) 17 pizzas did not have sausage

3 Pointers for the Exam(40 pts)

1. Logical Statement-2pts

2. Truth Values of Compound Statements- 5pts

3. Truth Value Problem-2pts

4. Truth Table, Tautology, Contradiction and Contingency-5pts

5. Reasoning(Inductive and Deductive)-5pts

8
 6. Counterexample-2pts

7. Problem Solving(Same type as in the examples above)-10 pts(2 items)

8. Definitions and Concepts(Please familiarize the concepts and defns)- 9 pts

I’ll give bonus points(maybe 3 pts) on some personal real life applications of math on the first
lesson.
Because of time constrain, I haven’t reviewed all the details of this material. If there are some
ambiguous parts, please notify me. Just study this diligently. God Bless...