Chapter 1 Notation & terminology Flashcards
1.1 Sets
What is a set?
A set is a collection of objects.
1.1 Sets
What is an element?
element of a set?
Each object in a set is said to be an element of the set.
1.1 Sets
What do round brackets and square brackets indicate in a set?
for example [0,10)
A round bracket indicases that the endpoint is not included in the set.
A square bracket indicates that the endpoint is included in the set.
[0,10) is the set of values that 0 0,10) is the set of values of x such
1.1 Sets
What does ∅ mean?
The empty set, ie. the set that contains no elements
1.1 Sets
What does ∈ mean?
Is a member of the set
if S is the set of even numbers, then 2∈S
1.1 Sets
What does ∉ mean?
Is not a member of the set
if S is the set of even numbers, then 13∉S
1.1 Sets
What does ⊂ mean?
Is a subset of
C⊂D if every element of the set C is also an element of the set D
1.1 Sets
What does ⊄ mean?
Is not a subset of
C⊄D means that the set C contains at least one element that is not an el
1.1 Sets
What does ∪ mean?
Union (Or)
C∪D is the set of elements contained in set C or set D or both
1.1 Sets
What does ∩ mean?
Intersection (And)
C ∩ D is the set of elements in set C that are also elements of set D
1.1 Sets
What does Ā mean?
Complement of set A
Ā is the set of elements that do not belong to set A
1.1 Sets
What does A’ mean?
Alternative notation for the complement of set A
complement being the set of elements that do not belong to set A
1.1 Sets
What does ℕ mean?
The set of natural numbers
ℕ = {1,2,3,4,…]
1.1 Sets
Should 0 be included in ℕ?
We use the convention that 0 is not a member of this set
Some mathematicians do define the set of natural number to include 0
1.1 Sets
What does ℤ mean?
The set of intergers
ℤ = {…,-3,-2,-1,0,1,2,3,…}
1.1 Sets
What does ℚ mean?
The set of rational number (fractions)
the set numbers of the form p/q, where p,q ∈ ℤ and q ≠ 0
1.1 Sets
What does ℝ mean?
The set of real numbers
the set of all number between - ∞ and ∞
1.1 Sets
What does the superscripts + or - mean in situations such as ℤ+ or ℤ-
In the example the + and - should be superscript
the + or - refers to the positive or negative numbers
with in the set respectivally
ℤ = {…,-3,-2,-1,0} = ℕ
1.2 Logic & Proofs
What does ∀ mean?
For all
(Xsquared) 1> ∀>x 1
1.2 Logic & Proofs
What does ∃ mean?
There exists
∃ x ∈ ℝ such that x+1= 5
1.2 Logic & Proofs
What does ∄ mean?
There does not exist
∄ x ∈ ℕ such that xsquared = 2
1.2 Logic & Proofs
What does : mean?
Such that
ℝ−= {x : - ∞< X<0}
1.2 Logic & Proofs
What does st mean?
Such that
∃x∈ ℝ st x + 1 = 5
1.2 Logic & Proofs
What does ⇒ mean?
Implies
x = -2 ⇒ xsquared = 4
1.2 Logic & Proofs
What does ⇐⇒ mean?
Implies and is implied by (equlivent to)
X = 0 ⇐⇒ xcubed = 0
1.2 Logic & Proofs
What does iff mean?
If and only if (same meaning as ⇐⇒)
n is even iff n/2 ∈ℤ
1.2 Logic & Proofs
What does → mean?
Tends to (or approaches)
1/x → 0 as x → ∞
1.2 Logic & Proofs
What does the superscript + mean in situations such as
x→1+
the + in the example should be superscript
X is approaching 1 from above - but is always slightly greater than 1
1/x → 0+ as x → ∞
Writting x → 1 without a super script + or - means x is approaching 1 from either direction
1.2 Logic & Proofs
What does the superscript - mean in situations such as
x→1-
the - in the example should be superscript
X is approaching 1 from below - is always slightly less than 1
1/x → 0- as x → -∞
Writting x → 1 without a super script + or - means x is approaching 1 from either direction
1.2 Logic & Proofs
How do we define Necessary?
If A is necessary for B,
then B ⇒ A,
B is true only if A is true
A = the number x is divisable by 5, B = the integer x ends in 5
1.2 Logic & Proofs
How do we define Sufficient?
If A is sufficient for B,
then A ⇒ B,
B is true if A is true
A = the integer x ends in 5, B = the number x is divisable by 5
1.2 Logic & Proofs
How do we define Necessary & Sufficient?
If A is necessary and sufficient for B,
then A ⇐⇒ B,
A and B are equlivent statments
A = the interger x is divisible by 2, B = the number x is even
1.3 Mathematical Constants
What is the mathmatical constant e?
Eulers Number
e is the natural language of growth
On a graph e to the power of x at any point has the same value, gradient & area under the curve
eg. interest on £1 every moment (1+1/n) to power of n → e as n → ∞
The value of e is: 2.718281828…
1.4 Greek letters
What does the greek letter α mean, and how is it most often used?
alpha = parameter
1.4 Greek letters
What does the greek letter β mean, and how is it most often used?
lower case beta = parameter
or upper case = B = beta function
1.4 Greek letters
What does the greek letter B mean, and how is it most often used?
upper case beta = beta function
or lower case = β = parameter
1.4 Greek letters
What does the greek letter γ mean, and how is it most often used?
lower case gamma = parameter
or upper case = Γ = gamma function
1.4 Greek letters
What does the greek letter Γ mean, and how is it most often used?
upper case gamma = gamma function
or lower case = γ = parameter
1.4 Greek letters
What does the greek letter δ mean, and how is it most often used?
lower case delta = small change
or upper case = Δ = difference
1.4 Greek letters
What does the greek letter Δ mean, and how is it most often used?
upper case delta = difference
or lower case = δ = small change
1.4 Greek letters
What does the greek letter ε mean, and how is it most often used?
epsilon = small quantity
1.4 Greek letters
What does the greek letter θ mean, and how is it most often used?
theta = parameter
1.4 Greek letters
What does the greek letter κ mean, and how is it most often used?
kappa = parameter
1.4 Greek letters
What does the greek letter λ mean, and how is it most often used?
lambda = parameter
1.4 Greek letters
What does the greek letter μ mean, and how is it most often used?
mu = mean, mortality rate
1.4 Greek letters
What does the greek letter ν mean, and how is it most often used?
nu = mortality rate when sick
1.4 Greek letters
What does the greek letter π mean, and how is it most often used?
lower case pi = 3.14
or upper case = Π = product
1.4 Greek letters
What does the greek letter Π mean, and how is it most often used?
upper case pi = product
or lower case = π = 3.14
1.4 Greek letters
What does the greek letter ρ mean, and how is it most often used?
rho = correlation coefficient, recovery rate
1.4 Greek letters
What does the greek letter σ mean, and how is it most often used?
lower case sigma = standard deviation, sickness rate
or upper case = Σ = sum
1.4 Greek letters
What does the greek letter Σ mean, and how is it most often used?
upper case sigma = sum
or lower case = σ = standard deviation, sickness rate
1.4 Greek letters
What does the greek letter τ mean, and how is it most often used?
tau = parameter
pronounced as in first syllable of tower (sometimes pronounced tall)
1.4 Greek letters
What does the greek letter φ mean, and how is it most often used?
lower case phi = probability density function of standard normal distrobution
upper case = Φ
1.4 Greek letters
What does the greek letter Φ mean, and how is it most often used?
upper case phi = cumulative distribution function of standard normal distribution
lower case = φ
1.4 Greek letters
What does the greek letter χ (χ²) mean, and how is it most often used?
chi =chi² = distribution
pronounced as first syllable of ‘Cairo’
1.4 Greek letters
What does the greek letter ψ mean, and how is it most often used?
psi = probability of ultimate ruin
1.4 Greek letters
What does the greek letter ω mean, and how is it most often used?
omega = limiting age in a life table
1.5 Expressions, equations, formulae, terms & factors
How would you define an ‘Expression’?
A mathematical expression is any combination of mathematical symbols, eg:
2+2,
1.09²
x + 2y,
2(b-c)
Usually expressions involve more than one symbol and many expressions include letters.
May not contain the ‘equal to’ sign or any type of inequality.
1.5 Expressions, equations, formulae, terms & factors
How would you define an ‘Equation’?
An equation is a statement concerning the equality of two expressions. Some examples of equations are given below:
2+2=4,
1.09² = 1.1881,
x+2y= -5
a(b-c)=ab-ac
In word processing packages the word ‘equation’ is often used more generally to mean anything containing mathematical symbols, which is not strictly correct
1.5 Expressions, equations, formulae, terms & factors
How would you define ‘Formulae’?
A formula is a special type of equation that shows the relationship between different quantities. For example, the formula for area of a triangle is as follows:
area =1/2×base height
A formula often uses letters to represent the variables. circumference of a circle is:
C=2πR
In this formula, C denotes circumference and r denotes radius.
1.5 Expressions, equations, formulae, terms & factors
How would you define Terms?
As in Terms, Factors, and Coefficients
A term is an element in an expression that is added or subtracted. For example, the terms in the expression ab-ac are ab and ac.
1.5 Expressions, equations, formulae, terms & factors
How would you define Factros?
As in Terms, Factors, and Coefficients
A factor is an element in an expression that is multiplied or divided. For example, the factors in the expression a(b-c) are a and b-c.
1.5 Expressions, equations, formulae, terms & factors
How would you define Coefficients?
As in Terms, Factors, and Coefficients
A numerical factor appearing in an expression, such as the 3 in 3x², is also called a coefficient.
1.6 Dimensions and units of measurement
What is a ‘Dimension’?
Dimensions are used to show what a numerical value actually represents.
pounds, meters, years, kilograms
Numbers/coefficients (inc π & e ) have no dimension & are dimensionless.
1.6 Dimensions and units of measurement
Can values with different dimensions be added or subtracted?
No, you can not add £2 and 5 years together. Where possable different units/dimensions need to be converted when adding, eg. £2 and $5 or 2m and 500cm
1.6 Dimensions and units of measurement
What happens when two values with the same dimension are divided?
The resulting values will be dimensionless.
10 years / by 5 years is 2 (not 2 years)
1.7 Conventions used in financial and actuarail mathematics
How would you commonaly write if the scientic notation (standard form) is: $6.2x10(po6)
£6.2m
1.7 Conventions used in financial and actuarail mathematics
What is the importance of using ‘precentage points’ and ‘basis points’?
To clearly diferantiage between adding/subtracting precentages, not multiplying them.
6% increased by 2 precentage points is 8%
6% increased by 2 precent is 6*1.02=6.12%
Basis points are used to comunicate this as 100th of a precentage point.
6% increased by 125bps is 7.25%
1.7 Conventions used in financial and actuarail mathematics
How are negative values shown in accounting?
In brackets
£(0.2m)
Is a negative of £200,000
1.7 Conventions used in financial and actuarail mathematics
What does this Δ mean?
Δ is used to denote a change in quantity. eg.
Δ profit = £534K means profit has risen by £534,000.
Δ profit = £(534K) means profit has fallen by £534,000.
Upper case Delta
1.8 Time Intervals
How are annual quaters defined?
Q1 = 1 Jan to 31 Mar
Q2 = 1 Apr to 30 Jun
Q3 = 1 Jul to 30 Sep
Q4 = 1 Oct to 31 Dec
Normally sufficiently accurate to assume a quarter is exactly 1/4 year
1.8 Time Intervals
When is the tax year (fiscal year) in the UK?
6 April to 5 April
So, for example, the 2014/15 tax year is the period from 6 April 2014 to 5 April 2015 (both days inclusive).
Tax years differ between countries.
1.8 Time Intervals
How is a year represented in actuarial notation?
using a right-anble symbol
5 years is represented by 5⌉
(but with the top of the box covering the top of the 5)
1.9 Glossary
A Life
Basic actuarial terminology
Just means a person
1.9 Glossary
A First-Class Life
Basic actuarial terminology
A person in perfect health
1.9 Glossary
An Impaired Life
Basic actuarial terminology
A person NOT in perfect health
1.9 Glossary
Immediate
Basic actuarial terminology
Within the next year
eg. an immediate pension would make the first payment at some time during the commingyear but not necessarily at the start of that year
Opposite of Deferred
1.9 Glossary
Deferred
Basic actuarial terminology
In the future (not within the next year)
eg. a deferred pension would normally start making payments a number of years in the future
Opposite of Immediate
1.9 Glossary
Level
Basic actuarial terminology
Constant
eg. level payments are for the same amount each time
1.9 Glossary
Net Pay/Payment
Basic actuarial terminology
Where something has been deducted.
Net subs = sub - IPT
Net pay = salary - tax, controbutions, study loan
Ask - what has been deducted?
1.9 Glossary
Gross Pay/Payment
Basic actuarial terminology
Total value without deductions.
Ask - does anything need to be deducted?
1.9 Glossary
Life Office/Office
Basic actuarial terminology
Just means insurance company
1.9 Glossary
Outgo
Basic actuarial terminology
amounts that are going out
opposite of income
1.9 Glossary
Income
Basic actuarial terminology
amounts that are coming in
opposite of outgo
1.9 Glossary
Payable
Basic actuarial terminology
must be paid
not may be paid
1.9 Glossary
Secular
Basic actuarial terminology
in relation to time measured by reference to the calender
example, the statement ‘Mortality can be expected to improve over an individual’s lifetime because of secular effects’, means that in the future people are likely to live longer than in previous generations, as over time, there are likely to be medical advances.
1.9 Glossary
Duration
Basic actuarial terminology
in relation between two points in time
eg the time since you were born or since you took out your life insurance policy
1.9 Glossary
Stochastic
Basic actuarial terminology
allowing for random variation over time
opposite of Deterministic
1.9 Glossary
Deterministic
Basic actuarial terminology
not allowing for random variation over time
1.9 Glossary
Per Annum
Basic actuarial terminology
per year
1.9 Glossary
pa
pm
pcm
pq
Basic actuarial terminology
per annum
per month
per calcander year
per quarter
1.9 Glossary
Pro Rata
Basic actuarial terminology
in proportion
1.9 Glossary
Vice Versa
Basic actuarial terminology
the other way around
1.9 Glossary
e.g.
Basic actuarial terminology
for example
1.9 Glossary
i.e.
Basic actuarial terminology
that is
1.9 Glossary
c.f.
Basic actuarial terminology
compare
eg. using the approximation, I got £73.98 (cf £74.02 when calculated accurately)
