Chapter 1 Notation & terminology Flashcards

1
Q

1.1 Sets

What is a set?

A

A set is a collection of objects.

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2
Q

1.1 Sets

What is an element?

element of a set?

A

Each object in a set is said to be an element of the set.

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3
Q

1.1 Sets

What do round brackets and square brackets indicate in a set?

for example [0,10)

A

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

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4
Q

1.1 Sets

What does ∅ mean?

A

The empty set, ie. the set that contains no elements

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5
Q

1.1 Sets

What does ∈ mean?

A

Is a member of the set

if S is the set of even numbers, then 2∈S

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6
Q

1.1 Sets

What does ∉ mean?

A

Is not a member of the set

if S is the set of even numbers, then 13∉S

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7
Q

1.1 Sets

What does ⊂ mean?

A

Is a subset of

C⊂D if every element of the set C is also an element of the set D

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8
Q

1.1 Sets

What does ⊄ mean?

A

Is not a subset of

C⊄D means that the set C contains at least one element that is not an el

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9
Q

1.1 Sets

What does ∪ mean?

A

Union (Or)

C∪D is the set of elements contained in set C or set D or both

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10
Q

1.1 Sets

What does ∩ mean?

A

Intersection (And)

C ∩ D is the set of elements in set C that are also elements of set D

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11
Q

1.1 Sets

What does Ā mean?

A

Complement of set A

Ā is the set of elements that do not belong to set A

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12
Q

1.1 Sets

What does A’ mean?

A

Alternative notation for the complement of set A

complement being the set of elements that do not belong to set A

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13
Q

1.1 Sets

What does ℕ mean?

A

The set of natural numbers

ℕ = {1,2,3,4,…]

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14
Q

1.1 Sets

Should 0 be included in ℕ?

A

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

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15
Q

1.1 Sets

What does ℤ mean?

A

The set of intergers

ℤ = {…,-3,-2,-1,0,1,2,3,…}

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16
Q

1.1 Sets

What does ℚ mean?

A

The set of rational number (fractions)

the set numbers of the form p/q, where p,q ∈ ℤ and q ≠ 0

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17
Q

1.1 Sets

What does ℝ mean?

A

The set of real numbers

the set of all number between - ∞ and ∞

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18
Q

1.1 Sets

What does the superscripts + or - mean in situations such as ℤ+ or ℤ-

In the example the + and - should be superscript

A

the + or - refers to the positive or negative numbers
with in the set respectivally

ℤ = {…,-3,-2,-1,0} = ℕ

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19
Q

1.2 Logic & Proofs

What does ∀ mean?

A

For all

(Xsquared) 1> ∀>x 1

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20
Q

1.2 Logic & Proofs

What does ∃ mean?

A

There exists

∃ x ∈ ℝ such that x+1= 5

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21
Q

1.2 Logic & Proofs

What does ∄ mean?

A

There does not exist

∄ x ∈ ℕ such that xsquared = 2

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22
Q

1.2 Logic & Proofs

What does : mean?

A

Such that

ℝ−= {x : - ∞< X<0}

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23
Q

1.2 Logic & Proofs

What does st mean?

A

Such that

∃x∈ ℝ st x + 1 = 5

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24
Q

1.2 Logic & Proofs

What does ⇒ mean?

A

Implies

x = -2 ⇒ xsquared = 4

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# 1.2 Logic & Proofs What does ⇐⇒ mean?
Implies and is implied by (equlivent to) | X = 0 ⇐⇒ xcubed = 0
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# 1.2 Logic & Proofs What does iff mean?
If and only if (same meaning as ⇐⇒) | n is even iff n/2 ∈ℤ
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# 1.2 Logic & Proofs What does → mean?
Tends to (or approaches) | 1/x → 0 as x → ∞
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# 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 → ∞ ## Footnote Writting x → 1 without a super script + or - means x is approaching 1 from either direction
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# 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 → -∞ ## Footnote Writting x → 1 without a super script + or - means x is approaching 1 from either direction
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# 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
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# 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
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# 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
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# 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 → ∞ ## Footnote The value of e is: 2.718281828...
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# 1.4 Greek letters What does the greek letter α mean, and how is it most often used?
alpha = parameter
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# 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
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# 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
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# 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
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# 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
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# 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
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# 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
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# 1.4 Greek letters What does the greek letter ε mean, and how is it most often used?
epsilon = small quantity
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# 1.4 Greek letters What does the greek letter θ mean, and how is it most often used?
theta = parameter
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# 1.4 Greek letters What does the greek letter κ mean, and how is it most often used?
kappa = parameter
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# 1.4 Greek letters What does the greek letter λ mean, and how is it most often used?
lambda = parameter
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# 1.4 Greek letters What does the greek letter μ mean, and how is it most often used?
mu = mean, mortality rate
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# 1.4 Greek letters What does the greek letter ν mean, and how is it most often used?
nu = mortality rate when sick
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# 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
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# 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
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# 1.4 Greek letters What does the greek letter ρ mean, and how is it most often used?
rho = correlation coefficient, recovery rate
50
# 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
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# 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
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# 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)
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# 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
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# 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
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# 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’
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# 1.4 Greek letters What does the greek letter ψ mean, and how is it most often used?
psi = probability of ultimate ruin
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# 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
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# 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.
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# 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
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# 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.
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# 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.
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# 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.
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# 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.
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# 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.
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# 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
66
# 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)
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# 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
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# 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%
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# 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
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# 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
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# 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
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# 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.
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# 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)
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# 1.9 Glossary A Life | Basic actuarial terminology
Just means a person
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# 1.9 Glossary A First-Class Life | Basic actuarial terminology
A person in perfect health
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# 1.9 Glossary An Impaired Life | Basic actuarial terminology
A person NOT in perfect health
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# 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
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# 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
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# 1.9 Glossary Level | Basic actuarial terminology
Constant eg. level payments are for the same amount each time
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# 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?
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# 1.9 Glossary Gross Pay/Payment | Basic actuarial terminology
Total value without deductions. | Ask - does anything need to be deducted?
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# 1.9 Glossary Life Office/Office | Basic actuarial terminology
Just means insurance company
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# 1.9 Glossary Outgo | Basic actuarial terminology
amounts that are going out | opposite of income
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# 1.9 Glossary Income | Basic actuarial terminology
amounts that are coming in | opposite of outgo
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# 1.9 Glossary Payable | Basic actuarial terminology
must be paid | not may be paid
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# 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.
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# 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
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# 1.9 Glossary Stochastic | Basic actuarial terminology
allowing for random variation over time | opposite of Deterministic
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# 1.9 Glossary Deterministic | Basic actuarial terminology
not allowing for random variation over time
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# 1.9 Glossary Per Annum | Basic actuarial terminology
per year
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# 1.9 Glossary pa pm pcm pq | Basic actuarial terminology
per annum per month per calcander year per quarter
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# 1.9 Glossary Pro Rata | Basic actuarial terminology
in proportion
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# 1.9 Glossary Vice Versa | Basic actuarial terminology
the other way around
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# 1.9 Glossary e.g. | Basic actuarial terminology
for example
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# 1.9 Glossary i.e. | Basic actuarial terminology
that is
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# 1.9 Glossary c.f. | Basic actuarial terminology
compare eg. using the approximation, I got £73.98 (cf £74.02 when calculated accurately)