Illustrating

Simple and
Compound
Interest
 Terms
Lender or creditor
 Person (or institution) who invests the
money or makes the funds available
Borrower or debtor
 Person (or institution) who owes the
money or avails of the funds from the
lender
Origin or loan date
 Date on which money is received by
the borrower
 Terms
Repayment date or maturity date
 Date on which the money borrowed or
loan is to be completely repaid
Time or Term (t)
 amount of time in years the money is
borrowed or invested; length of time
between the origin and maturity dates
Principal (P)
 Amount of money borrowed or invested
on the origin date
 Terms
Rate (r)
 Annual rate, usually in percent,
charged by the lender, or rate increase
of the investment.
Interest
 Amount paid or earned for the used of
money
Simple interest (Is)
 Interest that is computed on the
principal and then added to it.
 Terms
Compound Interest (Ic)
 Interest is computed on the principal
and also on the accumulated past
interests
Maturity value or future value (F)
 Amount after t years that the lender
receives from the borrower on the
maturity date
 Example
Problem Solving: Due to COVID-19 pandemic Mrs.
Batara a female resident of Brgy. Sinait
somewhere in Isabela Province thinks of a
business that can provide for her needs as well
as the need of her neighbors so she can be of
help even in this trying time.

Since she doesn’t have money on hand, she

decided to borrow from a bank as the start-up
capital of ₱50,000.00 at 7% simple interest rate
payable within 5 years. Compute for the interest
yield.
Solution
 Example
Problem Solving: Due to COVID-19 pandemic
Mrs. Batara a female resident of Brgy. Sinait
somewhere in Isabela Province thinks of a
business that can provide for her needs as well
as the need of her neighbors so she can be of
help even in this trying time.

Since she doesn’t have money on hand, she

decided to borrow from a bank with a start-up
capital of ₱50,000.00 at 7% interest rate
compounded annually and payable within 5
years. Compute for the interest yield.
Solution
 Simple Interest
Annual Simple Interest
Is = Prt
where
Is - simple interest
P – principal, or the amount invested or borrowed
r – simple interest rate
t – term or time in years
 Example 1
A bank offers 0.25% annual simple
interest rate for a particular deposit.
ow much interest will earn if 1 million
pesos is deposited in this savings
account for 1 year?
Given:
P- 1, 000,000 r – 0.25% or
0.0025
t – 1 year Is - ?
 Example 2

How much interest is charged when

50,000 is borrowed for 9 months at an
annual interest rate of 10%?

Given:
P- 50, 000 r – 10% or
0.10
t – 9 months yr Is - ?
 Activity
Complete the table bellow by finding the unknown
Principal Rate (r) Time (t) Interest
(P)
(a) 2.5% 4 1, 500
36, 000 (b) 1.5 4860
250, 000 0.5% (c) 275
500, 000 12. 5% 10 (d)
 Maturity (Future)
Value
F = P(1+rt)
where
F – maturity of future value
P – principal
r – interest rate
t – term/ time in years
 Example 1
1. Find the maturity value if 1 million
pesos is deposited in a bank at an
annual simple interest rate of 0.25%
after 5 years?

2. Find the maturity value if 25000

pesos is deposited in a bank at an
annual simple interest rate of 3% after 4
years?
 Compound Interest
Maturity (Future) Value and Compound
Interest
F = P(1+r) t
where
F – maturity of future value
P – principal
r – interest rate
t – term/ time in years

The compound interest Ic is given by Ic =

1. Find the maturity value and the
compound interest if 10,000 is
compounded annually at an interest
rate of 2% in 5 years.

2. Find the maturity value and interest

if 50, 000 is invested at 5%
compounded annually for 8 years.
 Present Value (P) at
Compound Interest
P=
where
P – principal or present value
F – maturity of future value
r – interest rate
t – term/ time in years
1. Find the present value of 50, 000 due
in 4 years if money is invested at
12% compounded annually?

2. How much money should a student

place in a time deposit in a bank that
pays 1.1% compounded annually so
that he will have 200, 000 after 6
years?
 Compounding More than Once a
Year
Definition of Terms
Frequency of Conversion
-number of conversion periods in one year
Conversion of Interest Period
- Time between successive conversions
of interest
Total Number of Conversion periods n
n = mt=(frequency of conversion)x(time in
Nominal rate (im)
-annual rate of interest
Rate (j) of interest for each
conversion period

j=
 Maturity Value (F), Compounding m times
a year
F =P () mt
where
F – maturity of future value
P – principal or present value
im - nominal rate of interest (annual rate)
m – frequency of conversion
t – term/ time in years
1. Find the maturity value and interest
if 10, 000 is deposited in a bank at
2% compounded quarterly for 5
years.

2. Find the maturity value and interest

if 10, 000 is deposited in a bank at
2% compounded monthly for 5
years.
Present Value (P) at Compound Interest

P=
where
P – principal or present value
F – maturity of future value
im - nominal rate of interest (annual rate)
m – frequency of conversion
t – term/ time in years
1. Find the present value of 50, 000 due
in 4 years if money is invested at
12% compounded semi-annually.

2. What is the present value of 25, 000

due in 2 years and 6 months if
money is worth 10% compounded
quarterly?
Activity:
1. Cian lends 45, 000 for 3 years at 5%
compounded semi-annually . Find the future
value and interest of this amount.
2. Tenten deposited 10, 000 in a bank which gives
1% compounded quarterly and let it stay there
for 5 years. Find the maturity value and interest.
3. How much should you set aside and invest in a
fund earning 9% compounded quarterly if you
want to accumulate 200, 000 in 3 years and 3
months?
4. How much should you deposit in a bank paying
2% compounded quarterly to accumulate an
amount of 80, 000 in 5 years and 9 months?
 Simple
Annuity
 Terms
Annuity
 A sequence of payments made at equal (fixed)
intervals or periods of time.
Annuities may be classified in different ways, as
According to follows.Annuities
duration
According to payment Simple Annuity- an annuity General Annuity- an annuity
interval and interest where the payment interval is where the payment interval is
period the same as the interest not the same as the interest
period period
According to time of Ordinary annuity- a type of Annuity Due- a type of
payment annuity in which the payments annuity in which the
are made at the end of each payments are made at
payment interval beginning of each payment
interval
According to duration Annuity certain- an annuity Contingent Annuity-
in which payments begin and annuity in which the
end at definite times payments extend over an
indefinite length of time
 Terms
Term of an annuity
 Time between the first payment interval and
last payment interval
Regular of periodic payment (R)
 The amount of each payment

Amount (Future Value) of an annuity (F)

 Sum of future values of all the payments to be
made during the entire term of the annuity
Present Value of an annuity (P)
 Sum of present values of all the payments to
be made during the entire term of the annuity
 Amount (Future Value) of
Ordinary Annuity
The future value F, of an ordinary annuity is given by

F=R
where
R – regular payment
j – interest rate per period
n – number of payments
1. Suppose Mrs. Batara would like to save
3,000 every months in a fund that gives 9%
compounded monthly. How much is the
amount or future value of her savings after
6 months?

2. In order to save for her high school

graduation, Gian decided to save 200 pesos
at the end of each month. If the bank pays
0.250% compounded monthly, how much
will her money be at the end of 6 years?
 Present Value of an Ordinary
Annuity
The present value P, of an ordinary annuity is given by

P=R
where
R – regular payment
j – interest rate per period
n – number of payments

The cash value or cash price of a purchase is equal to

the down payment (if there is any) plus the present value
of the installment payments.
1. Suppose Mrs. Remoto would like toknow the
present value of her monthly deposit of P3,000
when interest is 9%compounded monthly. How
much is the present value of her savings at the
end of 6 months?

2. Mr. Cadorna paid P200,000 as down payment

for a car. The remaining amount is to be settled
by paying P16,200 at the end of each month for
5 years. If interest is 10.5% compounded
monthly, what is the cash price of his car?
 Periodic payment R of an Annuity:
Periodic payment R can also be solved using the formula
for amount For present value P of an annuity

R = F/ ( )
R = P/( )
where
R is the regular payment;
P is the present value of an annuity
F is the future value of an annuity
j is the interest rate per period;
n is the number of payments
1.Paolo borrowed P 100 000. He agrees to pay the
principal plus interest by paying an equal amount of
money each year for 3 years. What should be his
annual payment if interest is 8% compounded
annually?

2.Mr. Reccion would like to save P500,000 for his son’s

college education. How much should he deposit in a
savings account every 6 months for 12 years if interest
is at 1%compounded semi-annually?
1. In order to save for her high school graduation, Gian
decided to save 200 pesos at the end of each month. If the
bank pays 0.250% compounded monthly, how much will
her money be at the end of 6 years?
2. Mr. Cadorna paid P200,000 as down payment for a car.
The remaining amount is to be settled by paying P16,200
at the end of each month for 5 years. If interest is 10.5%
compounded monthly, what is the cash price of his car?
3. Mr. Reccion would like to save P500,000 for his son’s
college education. How much should he deposit in a
savings account every 6 months for 12 years if interest is
at 1%compounded semi-annually?
A. Find the future value F of the following
ordinary annuities.
1. Monthly payments of P3,000 for 4 years with
interest rate of 3%compounded monthly
2. Quarterly payment of P5,000 for 10 years with
interest rate of 2%compounded quarterly
3. Semi-annual payments of P12, 500 with
interest rate of 10.5%compounded semi-
annually for 6 years
4. Annual payments of P105,000 with
interest rate of 12% compounded annually for
5 years
B. Find the present value P of the following
ordinary annuities.
6. Monthly payments of P2,000 for 5 years with
interest rate of 12%compounded monthly
7. Quarterly payment of P15,000 for 10 years
with interest rate of 8%compounded quarterly
8. Semi-annual payments of P20,500 with
interest rate of 8.5%compounded semi-annually
for 3 years
9. Annual payments of P150,000 with interest
rate of 8% compounded annually for 10 years
10. Daily payments of P54 for 30 days with
C. Find the periodic payments of the following
ordinary annuities.
11. Monthly payment of the future value of
P50,000 for 1 year with an interest rate of 10%
compounded monthly
12. Quarterly payment of an accumulated amount
of P80,000 for 2 years with interest rate of 8%
compounded quarterly
13. Payment every six months for the present
value of P100,000 for 2years with an interest rate
of 12% compounded semi-annually
14. Annual payment of the loan P800,000 for 5
years with an interest rate of 9% compounded
General
Annuity
General Annuity
an annuity where the length of the payment interval is
not the same as the length of the interest
compounding period
General Ordinary Annuity
a general annuity in which the periodic payment
ismade at the end of the payment interval

Examples of General annuity

1. Monthly installment payment of a car, lot, or house
with an interest rate that is compounded annually
2. Paying a debt semi-annually when the interest
is compounded monthly
Future and Present Value of a General Ordinary
Annuity
The future value F and present value P of a
general ordinary annuity is given by

F=R P=R
Where: R is the regular payment;
j is the equivalent interest rate per payment interval
converted from the interest rate per period;
and
n is the number of payments.
1. Cris started to deposit P1,000 monthly
in a fund that pays 6%compounded
quarterly. How much will be in the fund
after 15 years?
2. Ken borrowed an amount of money
from Kat. He agrees to pay the principal
plus interest by paying P38,973.76 each
year for 3 years. How much money did
he borrow if interest is 8% compounded
quarterly?
1. Teacher Kaye is saving P2,000 every
month by depositing it in a bank that
gives an interest of 1% compounded
quarterly. How much will she save in 5
years?
2. Vladimir purchased a new car for
P99,000 down payment and P15,000
every month. If the payments are based
on 7% compounded quarterly for 5
years, what is the total cash price of his
car?
Deferred
Annuity
Deferred Annuity- an annuity
that does not begin until a given
time interval has passed
Period of Deferral – time
between the purchase of an
annuity and the start of the
payments for the deferred annuity
Present Value of a Deferred Annuity
Present value P of a Deferred annuity is given by

P=

Where: R is the regular payment;

j is the equivalent interest rate per payment interval
converted from the interest rate per period;
and
n is the number of payments.
k is the number of conversion periods in the deferral
1. A credit card company offers a deferred payment
option for the purchase of any appliance. Rose
plans to buy a smart television set with monthly
payments of 4 000 for 2 years. The payments will
start at the end of 3 months. How much is the cash
price of the TV set if the interest rate is 10%
compounded monthly?

2. Melwin availed of a loan from the bank that gave

him an option to pay 20 000 monthly for 2 years.
The first payment is due after 4 months. How much
is the present value of the loan if the interest rate is
10% compounded monthly?
Stocks and
Bonds
STOCKS
Some corporations may raise money for their expansion
by issuing stocks. Stocks are shares in the ownership of
the company. Owners of stocks may be considered as
part owners of the company. There are two types of
stocks: common stock and preferred stock. Both will
receive dividends or share of earnings of the company.
Dividends are paid first to preferred shareholders.
Stocks can be bought or sold at its current price called
the market value. When a person buys some shares, the
person receives a certificate with the corporation's name,
owner’s name, number of shares and par value per
share.
BONDS
Bonds are interest bearing security which promises to pay
amount of money on a certain maturity date as stated in
the bond certificate. Unlike the stockholders, bondholders are
lenders to the institution which may be a government or private
company. Some bond issuers are the national government,
government agencies, government owned and controlled
corporations, non-bank corporations, banks and multilateral
agencies. Bondholders do not vote in the institution’s annual
meeting but the first to claim in the institution’s earnings. On
the maturity date, the bondholders will receive the face amount
of the bond. Aside from the face amount due on the maturity
date, the bondholders may receive coupons
(payments/interests), usually done semi-annually, depending on
the coupon rate stated in the bond certificate.
 5 1495 420 1 Number of
Shares
Certificate Corporatio
Number 2 n Issuing
ALJUN D. CADORNA the
Certificate

Share
3
Holder or
Stockholde
r
PAR
VALUE
1000 4 Par Value
Comparison of Stocks and
Bonds Stocks Bonds
A form of equity financing or A form of debt financing, or raising
raising money by allowing money by borrowing from investors
investors to be part owners of the
company
Stock prices vary every day. These Investors are guaranteed interest
prices are reported in various payments and a return of their
media (newspaper, TV, internet, money at the maturity date.
etc.).
Investors can earn if the stock Investors still need to consider the
prices increase, but they can lose borrowers credit rating. Bonds
money if the stock prices decrease issued by the government pose
or worse, if the company goes less risk than those by companies
bankrupt because the government has
guaranteed funding (taxes) from
which it can pay its loans.
Stocks Bonds
High risk but with Lower risk but lower yield
possibility of higher
returns
Can be appropriate if the Can be appropriate for
investment is for the long retirees (because of the
term (10 years or more). guaranteed fixed income)
This can allow investors to or for those who need the
wait for stock prices to money soon (because they
increase if ever they go cannot afford to take a
low. chance at the stock
market)
Definition of Terms in Relation to
Stocks
Stocks – share in the ownership of a
company
Dividend – share in the company’s profit
Dividend Per Share – ratio of the
dividends to the number of shares
Stock Market – a place where stocks can
be bought or sold. The stock market in the
Philippines is governed by the Philippine
Stock Exchange (PSE)
Market Value – the current price of a stock at
which it can be sold
Stock Yield Ratio – ratio of the annual
dividend per share and the market value per
share. Also called current stock yield.
Par Value – the per share amount as stated on
the company certificate. Unlike market value, it
is determined by the company and remains
stable overtime
Example 1.
A certain financial institution declared a P30,000,000 dividend
for the common stocks. If there are a total of 700,000 shares of
common stock, how much is the dividend per share?
Given: Total Dividend = P30,000,000
Total Shares = 700,000
Find: Dividend per Share
Solution.

= 42.86
Therefore, the dividend per share is P42.86
Example 2.
A certain corporation declared a 3% dividend on
a stock with a par value of P500. Mrs Lingan
owns 200 shares of stock with a par value of
P500. How much is the dividend she received?

Given:
Dividend Percentage = 3%
Par Value = P500
Number of Shares = 200
Find: Dividend
Solution
Dividend =(Dividend Percentage)x(Par Value) x(No. of Shares)

= (0.03)(500)(200)= 3,000

Thus, the dividend is P3,000

 Example 3.
Corporation A, with a current market value of P52, gave a dividend
of P8per share for its common stock. Corporation B, with a current
market value of P95,gave a dividend of P12 per share. Use the
stock yield ratio to measure how much dividends shareholders
are getting in relation to the amount invested.
Solution. Solution.
Given: Corporation A: Given: Corporation B:
Dividend per share = P8 Dividend per share = P12
Market value = P52 Market value = P95
Find: stock yield ratio Find: stock yield ratio

Stock yield ratio = 0.1538 or Stock yield ratio = 0.1263 or

15.38% 12.63%
Definition of Terms in Relation to
Bonds
Bond – interest-bearing security which promises to pay
(1) a stated amount of money on the maturity date, and
(2) regular interest payments called coupons.
Coupon – periodic interest payment that the
bondholder receives during the time between purchase
date and maturity date; usually received semi-annually
Coupon Rate – the rate per coupon payment period;
denoted by r
Price of a Bond – the price of the bond at purchase
time; denoted by P
Par Value or Face Value
- the amount payable on the maturity date; denoted by
F.

If P = F, the bond is purchased at par

If P < F, the bond is purchased at a discount
If P > F, the bond is purchased at premium

Term of a Bond – fixed period of time (in years) at

which the bond is redeemable as stated in the bond
certificate; number of years from time of purchase to
maturity date.
Fair Price of a Bond – present value of all cash inflows
to the bondholder.
Example 4.
Determine the amount of the semi-annual coupon for a
bond with a face value of P300,000 that pays 10%, payable
semi-annually for its coupons.
Given: Face Value F = 300,000
Coupon rate r = 10%
Find: Amount of the semi-annual coupon

Solution.
Annual coupon amount: 300,000(0.10) = 30,000.
Semi-annual coupon amount: 30,000 = 15,000

Thus, the amount of the semi-annual coupon is P15,000.

Example 5.
Suppose that a bond has a face value of P100,000 and
its maturity date is 10 years from now. The coupon rate
is 5% payable semi-annually. Find the fair price of this
bond, assuming that the annual market rate is 4%.
Given: Coupon rate r = 5%, payable semi-annually
Face Value = 100,000
Time to maturity = 10 years
Number of periods = 2(10) = 20
Market rate = 4%
Solution: Solution:
Amount of semi-annual coupon: 100,000 () = 2500
 Theory of
Efficient Markets
Definition of Terms
Fundamental Analysis – analysis of
various public information (e.g., sales,
profits) about stock.
Technical Analysis – analysis of patterns
in historical prices of a stock.
Weak form of efficient Market Theory –
asserts that stock prices already
incorporate all past market trading data
and information (historical price
information) only.
Semi-strong form of Efficient Market
Theory – asserts that stock prices already
incorporate all publicly available
information only.

Strong form of Efficient Market Theory

– asserts that stock prices already
incorporate all information (public and
private).
BUSINESS
&
CONSUMER
LOANS
 Business Loan

Business Loan - referred as the

borrowed money from a bank or
other lending institutions/persons
that can be used to start a
business or to have a business
 Consumer Loan

Consumer Loan - referred as

the borrowed money from a bank
or other lending
institutions/persons that can be
used for personal or family
 Let’s Solve and Learn!

1. Mr. Cadorna borrowed ₱ 250,000

from a bank to purchase a residential
lot. The rate of interest of his loan is
7.5% annually the loan is to be paid for
2 years. How much is to be paid after 2
years?
Given: Solution:
Find: Future Amount
P=₱ 250,000
𝑖m = 7.5% 𝑜𝑟 0.075
(F)
F=P (1+j)n
j= = = 0.075
F=250,000(1+0.075)
n= (m)(t) = (1)(2)= 2
2 Thus, the amount to be
F= ₱288,906.25
paid by Mr. Cadorna
after 2 years is
₱288,906.25
 Let’s Solve and Learn!

2. A housing loan amounting to

₱870,000 requires a 20% down
payment. How much is the mortgage?
 Find: amount of loan or
mortgage
Given:
Down payment = down payment rate x
down payment cash price

rate= 20% or = 0.20 x ₱870,000

0.20 = ₱174,000
Amount of the Loan = Cash Price-Down
Cash price = Payment
₱870,000 The amount of the loan
= ₱870,000- or
₱174,000

mortgage is ₱696,000.
= ₱696,000
 It’s Your Turn!

Mr. Madamba borrowed ₱2,000,000 for

the expansion of his farm supply
business. The effective rate of interest
is 8%. The loan is to be repaid in full
after three years. How much is to be
paid after three years?
Given: Solution:
Find: Future Amount
P=₱ 2,000,000
𝑖m = 8% 𝑜𝑟 0.08
(F)
F=P (1+j)n
j= = = 0.08
F=2,000,000(1+0.08
n= (m)(t) = (1)(2)= 3
)
3 Thus, the amount to be
F= ₱ 2,519,424
paid after 3 years is
₱2,519,424
 It’s Your Turn!

If a 2- hectares land is to be sold for

₱2,000,000 and the lender requires
30% down payment, what is the
amount of the mortgage?
 Find: amount of loan or
Given: mortgage
Down payment = down payment rate x
down payment
cash price
rate= 30% or = 0.30 x ₱2,000,000

0.30 = ₱ 600,000
Cash price = Amount of the Loan = Cash Price-Down
Payment
₱2,000,000 The amount of the loan or
= ₱2,000,000 - ₱600,000
mortgage is ₱1,400,000.
= ₱ 1,400,000
Outstanding
Balance
Outstanding Balance
Any remaining debt at a specified
time.

BK =
1. Mrs. Se borrowed some money from a bank
that offers an interest rate of 12% compounded
monthly. Her monthly amortization for 5 years
is P11,122.22.How much is the outstanding
balance after the 12th payment?

2. Mr. Baldonado is considering to pay his

outstanding balance after 6 years of payment.
The original amount of the loan is P500,000
payable annually in 10years. If the interest rate
is 10% per annum and the regular payment
isP81,372.70 annually, how much is the
outstanding balance after the 6th payment?
3. Mrs. Tan got a business loan worth P800,000. She
promised to pay the loan semi-annually in 5 years. The
semi-annual payment is P103,603.66 if money is worth
10% converted semi-annually. How much is the
outstanding balance after the third payment

4. Mr. Bainto has a loan that is to be amortized by

paying monthly payments ofP3,200 for 1 year. After
paying for 6 months, he decided to pay off the loan.
How much of the 6th payment goes to pay the principal
if money is worth 12%compounded monthly?
5. Ms.Lachica got a car loan that requires a
monthly payment of P13,000 for 5years. She
plans to pay off the loan after paying for 3 years.
How much of the13th payment goes to pay the
principal if the interest rate is 10% compounded
monthly?
Propositions
A proposition is a declarative sentence that is
either true or false, but not both. If a proposition is
true, then its truth value is true, which is denoted
by T; otherwise, its truth value is false, which is
denoted by F.

Propositions are usually denoted by small letters.

For example, the proposition
p: Everyone should study logic

may be read as
p is the proposition “Everyone should study
Determine whether each of the following
statements is a proposition or not. If it is a
proposition, give its truth value.

p: Mindanao is an island in the Philippines.

q: Find a number which divides your age.
r: My seatmate will get a perfect score in the logic
exam.
s: Welcome to the Philippines!
t: 3+2 =5
u: is a rational function.
v: What is the domain of the function?
w: I am lying.
x: It is not the case that is a rational
number.
y : Either logic is fun and interesting, or it is
boring.
z: If you are a Grade 11 student, then you
are a Filipino.
A compound proposition is a proposition formed from
simpler proposition using logical connectors or some
combination of logical connectors.
Some logical connectors involving propositions p and/or
q may be expressed as follows:
not p
p and q
p or q
If p, then q
where <.> stands for some proposition. A proposition is
simple if it cannot be broken-down any further into other
component propositions.
Determine whether it is a simple or a compound
proposition. If it is a compound proposition,
identify the simple components.
a: It is not the case that is a rational number.

b: Either logic is fun and interesting, or it is boring.

c: If you study hard, Then you will get good grades.

d: If you are more than 60 years old, then you are

entitled to a Senior Citizen’s card, and if you are entitled
to a Senior Citizen’s card, then you are more than 60
years old.
 Logical
Operators
The negation of a proposition p is
denoted by
~p (“not p”)
and is defined through its truth table

p ~p
T F
F T
State the negation of the following
propositions.

u: is a polynomial function.

v: 2 is an odd number.

w:The tinikling is the most difficult dance.

x: Everyone in Visayas speaks Cebuano

The conjunction of the propositions p and
q is denoted by
p ^ q ( p and q)
and is defined through its truth table
p q p^q
T T T
T F F
F T F
F F F

The propositions p and q are called conjuncts. The

conjunction p ^ q is true only when both conjuncts p and
q are true, as shown in its truth table.
 Let p and q be the propositions
p: Angels exist.
q:
Express the following conjunctions in
English sentences or in symbols, as the
case may be.
1. p^q
2. p^(~q)
3. Angels do not exist and
4. While angels do not exist,
The disjunction of two propositions p and q is
denoted by
p v q (p or q)
and is defined through its truth table
p q p^q
T T T
T F T
F T T
F F F

The propositions p and q are called disjuncts. The above

truth table shows us that the disjunction p v q is false
only when both disjuncts p and q are false.
 Let p, q, and r be the following propositions:
p: Victor has a date with Liza.
q: Janree is sleeping.
r: Eumir is eating.

Express the following propositions in English sentences or

in symbols, as the case may be
1. p v q
2. q v (~r)
3. p v (q v r)
4. “Either Victor has a date with Liza or Janree is
sleeping, or Eumir is eating.”

5. “Either Victor has a date with Liza and Janree is

sleeping, or Eumir is eating.”

6. “Either Victor has a date with Liza, or Janree is

sleeping, and Eumir is eating.”

7. “Either Victor has a date with Liza and Janree is

sleeping, or Victor has a date with Liza and Eumir is
eating.”
The conditional of the propositions p and
q is denoted by
p → q ( If p, then q)
and is defined through its truth table
p q p→ q
T T T
T F F
F T T
F F T

The CONDITIONAL p → q may also be read as “p implies

q”. The proposition p is called the hypothesis, while the
proposition is called the conclusion.
Suppose that Geebee is a Grade 11 student. Consider
the following conditionals.

p1: If Geebee is in Grade 11, then she is a senior

high school student.

p2: If Geebee is in Grade 11, then she is working

as a lawyer.

p3: If Geebee has a degree in computer science,

then she believes in true love.
The biconditional of the propositions p
and q is denoted by
p ↔ q (p if and only if q)
and is defined through its truth table
p q p ↔ q
T T T
T F F
F T F
F F T

The proposition may also be written as “p iff q”. The

proposition p are the components of the biconditional.
Suppose that Geebee is a Grade 11 student. Let
us now consider the following biconditionals.

p1: Geebee is in Grade 11 if and only if she is a

senior high school student.

p2: Geebee is in Grade 11 if and only if she is

working as a lawyer.

p3: Geebee has a degree in Computer Science if

and only if she believes in true love.