MA170 Lab Notes
Text References: 1.5, 2.1 - 2.4
Simple Discount at a Discount Rate (1.5)
-Present (discounted) values are sometimes calculated using an annual simple discount rate d
d=
amount of discount for year
amount S on which the discount is given
-D
MA170 Lab Notes
Text References: 3.4, 3.5
Discounted Types of Simple Annuities (3.4)
In general, an annuity is a sequence of periodic payments (usually equal) made at equal intervals (payment intervals)
of time.
1. Simple Annuity: An annuity where the pay
MA170 Lab Notes
Text References: 1.1 - 1.4
Simple Interest (1.1)
1. Principal P - Initial amount of money (capital) invested or borrowed.
2. Rate of interest r - annual percentage earned on the original investment of P .
3. Amount of simple interest earne
MA170 Lab Notes
Text References: 6.4 - 6.6
Types of Bonds (6.2 - 6.4)
1. Bond redeemable at par:
A bond for which the redemption value C is the same as the value F (common). If not specified, assume that
a bond is redeemed at par.
2. Bond purchased at a p
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MA170 Lab 10 - Bonds II
Name:
Student Number:
Spring 2017
1. [15 marks] A $5000 callable bond with coupons at j4 = 8% matures on March 28, 2023 at par. The bond is callable on
March 28, 2018 at a 15% pre
MA170 Lab Notes
Text References: 2.7, 2.8, 2.10, App. 2, 3.1 - 3.3
Compound Interest at Changing Interest Rates (2.7)
When dealing with changing interest rate problems, the idea is to break the timeline up into multiple parts where
each part of the timeli
MA170 Lab Notes
Text References: 2.4 - 2.6
Accumulated and Discounted (Present) Values for a fractional period of time (2.4)
Method 1: Exact or Theoretical Method
- Compound interest is accrued for fractional periods using
n
S = P (1 + i) ,
where n = tm m
MA170 Lab Notes
Text References: 6.1 - 6.4
Bonds (6.1)
A written contract between the issuer (borrower) and the investor (lender) which promises to pay a stated amount
(or amounts) of money at some future date (or dates). Lender is paid interest on invest
MA170 Lab Notes
Text References: 4.4, 5.1, 5.2, 4.2
Annuities Where Payments Vary (4.4)
In the case where the payments of the annuity differ by a common ratio, both the present value and the future value
of the annuity result in a geometric progression. O
MA170 Lab Notes
Text References: 3.6, 4.1 4.3
Determining the Interest Rate (3.6)
Linear Interpolation:
NOTE: The following method is the same method as the one described in the textbook, only it uses slightly different
notation and looks a little differe
MA121 Lab 5 - Tutorial
Spring 2017
1. [4.3 #5(a)] Use induction on k to prove that for all nonnegative integers k and n,
n1
Hint: Use the theorem that nk = n1
k1 +
k .
Pk
j=0
n+j
j
Proof: By induction.
Base Case: let k = 0
P0
LS = j=0 n+j
= n0 =
j
n+1
= R
MA121 Lab 5 - Quiz
Name:
Student Number:
Spring 2017
1. [5 marks]
(a) Prove that if A and B are both non-empty sets, then A B = B A implies A = B.
Proof #1: We will prove the contrapositive.
Let A and B be non-empty sets, such that A 6= B.
Then there is a
Spring 2017
MA121 Lab 1
Tutorial Problem Set - Introduction to Sets
1. Which of the following sets are equal?
A = cfw_k Z : |k| < 2
B = cfw_k Z : k 3 = k
C = cfw_k Z : k 2 k
D = cfw_k Z : k 2 1
E = cfw_k Z : k (1, 1)
F = cfw_k Z : k [1, 1]
G = cfw_1, 0, 1
MA121 Lab 2 - Tutorial
Spring 2017
1. Consider the propositional statements given by P (Q R) and (P Q) (P R).
(a) Construct a truth table for the compound statements given above.
P
T
T
T
T
F
F
F
F
Q
T
T
F
F
T
T
F
F
R
T
F
T
F
T
F
T
F
QR
T
F
F
F
T
F
F
F
P (
MA121 Lab 4 - Tutorial
Spring 2017
1. (a) Prove that if A B = then A = or B = .
Proof:
Assume A 6= and B 6=
x A and y B
(x, y) A B
A B 6=
(b) State the proof method used in part (a).
or
Contrapositive
Contradiction if assumption A B 6= was made, final l
MA121 Lab 2 - Quiz
Name:
Student Number:
Spring 2017
1. [3 marks] Which of the following sets are equal? Show your work.
A = cfw_k Z : k 3 = |k|
= cfw_0, 1
B = cfw_k Z : k 2 k
= cfw_0, 1
C = cfw_k Z : k 2 < 1
= cfw_0
D = cfw_k Z : k (2, 2)
= cfw_1, 0, 1
E
MA121 Lab 2 - Quiz
Name:
Student Number:
Spring 2017
1. [3 marks] Which of the following sets are equal? Show your work.
A = cfw_k Z : |k| 2
= cfw_2, 1, 0, 1, 2
B = cfw_k Z : k 2 k
= cfw_0, 1
C = cfw_k Z : k 2 < 1
= cfw_0
D = cfw_k Z : k (2, 2)
= cfw_1, 0
MA121 Lab 4 - Quiz
Name:
Student Number:
Spring 2017
1. [5 marks] Prove (by induction) that for all real numbers a1 , a2 , . , an such that 0 ai 1, then
(1 a1 )(1 a2 ) (1 an ) 1 (a1 + a2 + + an ), where n 2.
Proof: By induction.
Base Case:
For a1 = x, a2
MA121 Lab 4 - Tutorial
Spring 2017
1. (a) Prove that if A B = then A = or B = .
(b) State the proof method used in part (a).
2. Determine the cardinality of each of the following, be sure to show your working out.
Given:
The universal set is all the natur
MA121 Lab 3 - Tutorial
Spring 2017
1. For a, b R, |ab| = |a|b|. Prove this proposition using cases.
2
2. Prove (by induction) that for all positive integers, xi , (x1 + . . . + xn ) > (x1 )2 + . . . + (xn )2 , where n > 1.
3. Write each statement using lo
MA121 Lab 3 - Tutorial
Spring 2017
1. For a, b R, |ab| = |a|b|. Prove this proposition using cases.
Proof: A real number is either negative or non-negative.
Case 1:
Assume a, b 0.
By definition |a| = a and |b| = b, so |a|b| = ab.
We also have that ab 0, s
MA121 Lab 3 - Quiz
Name:
Student Number:
Spring 2017
1. [7 marks] Consider the propositional statements given by P Q and Q P.
(a) Construct a truth table for the compound statements given above.
P
T
T
F
F
Q
T
F
T
F
PQ
T
F
T
T
P
F
F
T
T
Q
F
T
F
T
Q P
T
F
MA121 Lab 2 - Tutorial
Spring 2017
1. Consider the propositional statements given by P (Q R) and (P Q) (P R).
(a) Construct a truth table for the compound statements given above.
(b) What type of propositonal statement is P (Q R) (P Q) (P R), explain ?
(c
MA121 Lab 1 - Tutorial
Spring 2017
1. Which of the following sets are equal? Show your work.
A = cfw_k Z : |k| < 2
= cfw_1, 0, 1
B = cfw_k Z : k 3 = k
= cfw_1, 0, 1
C = cfw_k Z : k 2 k
= cfw_0, 1
D = cfw_k Z : k 2 1
= cfw_1, 0, 1
E = cfw_k Z : k (1, 1)
=
MA121 Lab 3 - Quiz
Name:
Student Number:
Spring 2017
1. [7 marks] Consider the propositional statements given by P Q and Q P.
(a) Construct a truth table for the compound statements given above.
P
T
T
F
F
Q
T
F
T
F
PQ
T
F
T
T
P
F
F
T
T
Q
F
T
F
T
Q P
T
F
MA121 Lab 5 - Tutorial
1. [4.3 #5(a)] Use induction on k to prove that for all nonnegative integers k and n,
n1
Hint: Use the theorem that nk = n1
k1 +
k .
Spring 2017
Pk
j=0
n+j
j
=
n+k+1
k
.
2. (a) Use Euclids Algorithm to find gcd(121, 2017).
(b) Find
MA 121
Introduction to Mathematical Proofs Week 10
5.4 Congruence's on
Congruent Modulo
Definition 5.4.1 The integer a is congruent (or equivalent) modulo n to the
integer b iff the difference between a and b is a multiple of n. That is, =
for some inte
MA 121
Introduction to Mathematical Proofs Week 11 and 12
6.2 Complex Numbers
Standard form and i
= 1
So we can solve equations like 2 + 1 = 0
The standard form is = + where , are both real numbers
Multiplication and Addition
If = + and = + then
+ = + +
MA 121
Introduction to Mathematical Proofs Week 2
2.2 Statements, Connectives and
Truth Tables
Statements
Definition 2.2.1 A statement or proposition is a declarative sentence (that is, an assertion of
claim) which is always either true (denoted T) of fal
MA 121
Introduction to Mathematical Proofs Week 6
3.1 Logic versus Set Theory
Let , be set and statements , as , Then
is the same as
is the same as
is the same as
Recall Sets
Theorem 1.5.3 For any three sets A,B and C we have
1.=
2.and
3.=
4.cfw_=c