Preliminary Mathematics for online MSc programmes in Data AnalyticsUnit 1: Mathematical notation, sets, functions, exponents and logarithms

Mathematical notation and symbols

Introduction

This introductory block is here to remind you of some important notations and conventions used in Mathematics and Statistics.

Numbers and common notations

  • The numbers 1 comma 2 comma 3 comma ellipsis are called natural numbers. These are denoted by double struck upper N (whereas the set double struck upper N 0 denotes all natural numbers including the number 0).

  • Integers are denoted by double struck upper Z and include negative numbers too: ellipsis comma negative 2 comma negative 1 comma 0 comma 1 comma 2 comma ellipsis

  • Numbers that can be expressed as a ratio of two integers (that is, of the form StartFraction a Over b EndFraction where a and b are integers, and b not equals 0) are said to be rational.

  • Numbers such as StartRoot 2 EndRoot comma pi comma e cannot be expressed as a ratio of integers; thus they are called irrational.

  • The set of real numbers includes both rational and irrational numbers and is denoted by double struck upper R.

  • The reciprocal of any number is found if we divide 1 by that number. For example, the reciprocal of 3 is one third and the reciprocal of one third is 3. Note that the old denominator has become the new numerator, and the old numerator has become the new denominator.

  • The absolute value of a number can be thought of as its distance from zero. This is denoted by vertical lines around the number. For example, StartAbsoluteValue 6 EndAbsoluteValue (read "the absolute value of 6") is 6, and StartAbsoluteValue negative 6 EndAbsoluteValue is 6 again.

  • The factorial of a non-negative integer number n is denoted by n factorial (read "n factorial") and is the product of all positive integers less than or equal to n. For example 4 factorial equals 4 dot 3 dot 2 dot 1 equals 24. We also define 0 factorial to be equal to 1.

Using symbols

Mathematics provides a very rich language for the communication of different concepts and ideas. In order to use this language it is of high importance to appreciate how symbols are used to represent physical quantities, and to understand the rules and conventions that have been developed to manipulate them.

The choice of which letters or symbols to use is up to the user, although it is helpful to choose letters that have some meaning in any particular context. For example, if we wish to choose a symbol to represent the temperature in a room we might choose the capital letter upper T. Usually the lowercase letter t is used to represent time. Since both time and temperature can vary we refer to t and upper T as variables. In a particular calculation some symbols represent fixed and unchanging quantities and we call these constants.

We often reserve the letters x, y and z to stand for variables and use the earlier letters of the alphabet, such as a, b and c, to represent constants. The Greek letter pi is used to represent the constant 3.14159 ellipsis which appears in the formula for the area of the circle. Other Greek letters are frequently used, and for reference the Greek alphabet is given below.

LetterUpper caseLower caseLetterUpper caseLower case
Alphaupper AalphaNuupper Nnu
Betaupper BbetaXinormal upper Xixi
Gammanormal upper GammagammaOmicronupper Oo
Deltanormal upper DeltadeltaPinormal upper Pipi
Epsilonupper Eepsilon or epsilonRhoupper Prho
Zetaupper ZzetaSigmanormal upper Sigmasigma
Etaupper HetaTauupper Ttau
Thetanormal upper Thetatheta or thetaUpsilonupper Yupsilon
Iotaupper IiotaPhinormal upper Phiphi or phi
Kappaupper KkappaChiupper Xxi
Lambdanormal upper LamdalamdaPsinormal upper Psichi
Muupper MmuOmeganormal upper Omegaomega

Mathematics is a very precise language and care must be taken to note the exact position of any symbol in relation to any other. If x and y are two symbols, then the quantities x y, x Superscript y and x Subscript y can all mean different things. In the expression x Superscript y, y is called a superscript while in the expression x Subscript y it is called a subscript.

  • If the letters x and y represent two numbers, then their sum is written as x plus y.

  • Subtracting y from x yields x minus y. This quantity is also called the difference of x and y.

  • The instruction to multiply x and y is written as x times y where usually the multiplication sign is omitted and we simply write x y. This quantity is called the product of x and y.

  • Note that x y is the same as y x. Because of this we say that multiplication is commutative.

  • Multiplication is also associative. When we multiply three quantities together, such as x times y times z, it doesn't matter whether we evaluate x times y first and then multiply the result by z, or evaluate y times z first and then multiply the result by x. In other words, left parenthesis x times y right parenthesis times z equals x times left parenthesis y times z right parenthesis.

  • The quantity StartFraction x Over y EndFraction (or x/y) means that x is divided by y. In the expression StartFraction x Over y EndFraction the top line is called the numerator and the bottom line is called the denominator. Division by 1 leaves any number unchanged (i.e. StartFraction x Over 1 EndFraction is simply x) while division by 0 is never allowed.

  • The equals sign, equals, is used in several different ways:

    • It can be used in equations. The left-hand side and right hand side of an equation are equal only when the variable involved takes specific values known as solutions of the equation. For example, in the equation x minus 10 equals 0, the variable is x and the left-hand side and right-hand side are equal when x has the value 10. If x has any other value the two sides are not equal.
    • It can be used in formulae. Physical quantities are often related through a formula. For example, the formula of the length, upper C, of the circumference of a circle expresses the relationship between the circumference of the circle and its radius r. It specifically states that upper C equals 2 pi r. When used in this way the equals sign expresses the fact that the quantity on the left is found by evaluating the expression on the right.
    • It can also be used in identities. At first sight an identity looks like an equation, except that is true for all values of the variable. For example, left parenthesis x minus 1 right parenthesis left parenthesis x plus 1 right parenthesis equals x squared minus 1 is true for all values of the variable x.
  • The sign not equals is read "is not equal to". For example it is correct to write 12 not equals 21.

  • The sigma summation notation (read "Sigma notation") provides a convenient way of writing long sums. The sum x 1 plus x 2 plus x 3 plus ellipsis plus x 20 is written using the capital Greek letter sigma, sigma summation, as sigma summation Underscript i equals 1 Overscript i equals 20 Endscripts x Subscript i.

  • The product notation (read "product notation") provides a convenient way of writing long products. The product x 1 times x 2 times x 3 times ellipsis times x 20 is written using the capital Greek letter Pi, product, as product Underscript i equals 1 Overscript i equals 20 Endscripts x Subscript i.

Inequalities

Given any two real numbers a and b, there are three mutually exclusive possibilities:

  • a greater than b (a is greater than b),

  • a less than b (a is less than b), or

  • a equals b (a is equal to b).

The inequality in the first two cases is said to be strict.

The case where "a is greater than or equal to b" is denoted by a greater than or equals b. Similarly, we have that a less than or equals b.

In these cases, the inequalities are said to be weak.

Some useful relations are:

  • If a greater than b and b greater than c; then a greater than c.

  • If a greater than b; then a plus c greater than b plus c for any c.

  • If a greater than b; then a c greater than b c for any positive c.

  • If a greater than b; then a c less than b c for any negative c.

Laws of indices

Indices or powers provide a convenient notation when we need to multiply a number by itself several times. the number 5 times 5 times 5 is written as 5 cubed and read "5 raised to the power of 3". Similarly we could have

8 times 8 times 8 times 8 equals 8 Superscript 4 Baseline comma left parenthesis negative 2 right parenthesis times left parenthesis negative 2 right parenthesis equals left parenthesis negative 2 right parenthesis squared comma z times z times z times z times z equals z Superscript 5 Baseline period More generally, in the expression x Superscript y, x is called the base and y is called the index or power.

There are a number of rules that enable us to manipulate expressions involving indices. These rules are known as the laws of indices and they occur so commonly that it is worthwhile to memorise them.

The laws of indices state:

  • a Superscript m Baseline times a Superscript n Baseline equals a Superscript m plus n (when multiplying two numbers that have the same base we just add their indices)

  • StartFraction a Superscript m Baseline Over a Superscript n Baseline EndFraction equals a Superscript m minus n (when dividing two numbers that have the same base we subtract their indices)

  • left parenthesis a Superscript m Baseline right parenthesis Superscript n Baseline equals a Superscript m n (if a number is raised to a power and the result itself is raised to a power, the two powers are multiplied together)

Note that in all the previous rules the base was the same throughout.

Two important results that can be derived from these laws are that:

  • a Superscript 0 Baseline equals 1 (any number raised to the power of 0 is 1), and

  • a Superscript 1 Baseline equals a (any number raised to the power of 1 is itself).

A generalisation of the third law states:

  • left parenthesis a Superscript m Baseline b Superscript k Baseline right parenthesis Superscript n Baseline equals a Superscript m n Baseline b Superscript n k (when two numbers, a Superscript m and b Superscript k, are multiplied together and they are raised to the same power, each number is raised to that power and they can then be multiplied together).
Negative indices

A number can be raised to a negative power. This is interpreted as raising the reciprocal number to the positive power. For example, 5 Superscript negative 2 Baseline equals left parenthesis one fifth right parenthesis squared equals StartFraction 1 squared Over 5 squared EndFraction equals one twenty fifth.

Generally, we have that a Superscript negative m Baseline equals StartFraction 1 Over a Superscript m Baseline EndFraction and a Superscript m Baseline equals StartFraction 1 Over a Superscript negative m Baseline EndFraction.

Fractional indices

Let's now consider the expression left parenthesis 16 Superscript 1 divided by 2 Baseline right parenthesis squared. Using the third law of indices we can write it as StartLayout 1st Row 1st Column left parenthesis 16 Superscript 1 divided by 2 Baseline right parenthesis squared 2nd Column equals 16 Superscript one half 2 Baseline 2nd Row 1st Column Blank 2nd Column equals 16 Superscript 1 Baseline 3rd Row 1st Column Blank 2nd Column equals 16 period EndLayout

So 16 Superscript 1 divided by 2 is a number that when it is raised to the power of 2 equals 16. That means that it could be 4 or negative 4. In other words 16 Superscript 1 divided by 2 is a square root of 16, that is StartRoot 16 EndRoot. There are always two square roots of a non-zero number, and we write 16 Superscript 1 divided by 2 Baseline equals plus or minus 4.

Similarly, we have that StartLayout 1st Row 1st Column left parenthesis 8 Superscript 1 divided by 3 Baseline right parenthesis cubed 2nd Column equals 8 Superscript one third 3 Baseline 2nd Row 1st Column Blank 2nd Column equals 8 Superscript 1 Baseline 3rd Row 1st Column Blank 2nd Column equals 8 comma EndLayout

so that 8 Superscript 1 divided by 3 is a number that when it is raised to the power of 3 equals 8. Thus 8 Superscript 1 divided by 3 is the cubic root of 8, that is RootIndex 3 StartRoot 8 EndRoot which is equal to 2. Each number has only one cubic root.

Generally, we have that x Superscript StartFraction 1 Over n EndFraction is the n-th root of x, that is defined as RootIndex n StartRoot x EndRoot. The generalisation of the third law of indices states that left parenthesis a Superscript m Baseline b Superscript k Baseline right parenthesis Superscript n Baseline equals a Superscript m n Baseline b Superscript n k. By taking m equals k equals one half and n equals 1 we have that StartRoot a b EndRoot equals StartRoot a EndRoot StartRoot b EndRoot.

Polynomial expressions

An important group of mathematical expressions that use indices are known as polynomial expressions. Examples of polynomials are 5 x cubed minus 3 x squared plus 10 comma 11 minus 5 x Superscript 5 Baseline plus 7 x comma y minus y cubed period Notice that they are all constructed using non-negative whole-number powers of the variable. Recall that x Superscript 0 Baseline equals 1 and so the number 10 appearing in the first expression can be thought of 10 x Superscript 0.

A polynomial expression takes the form a 0 plus a 1 x plus a 2 x squared plus a 3 x cubed plus ellipsis where a 0 comma a 1 comma a 2 comma a 3 are all constants called the coefficients of the polynomial. The number a 0 is also called the constant term. The highest power in a polynomial is called the degree of the polynomial. Polynomials with degree 3, 2, 1 and 0 are known as cubic, quadratic, linear and constant respectively.

Tasks

Task 1

Write out explicitly what is meant by the following:

(a) sigma summation Underscript i equals 1 Overscript i equals 6 Endscripts k Superscript i

(b) sigma summation Underscript i equals 1 Overscript i equals 6 Endscripts i Superscript k

(c) sigma summation Underscript i equals 1 Overscript i equals 6 Endscripts left parenthesis i plus 1 right parenthesis Superscript k

(d) sigma summation Underscript i equals 1 Overscript i equals 6 Endscripts 2

(e) product Underscript i equals 1 Overscript i equals 6 Endscripts k Superscript i

(f) product Underscript i equals 1 Overscript i equals 6 Endscripts 2

Show answer

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VideoVideo model answers for part (a)Duration0:35VideoVideo model answers for part (c)Duration1:01

(a) k Superscript 1 Baseline plus k squared plus k cubed plus k Superscript 4 Baseline plus k Superscript 5 Baseline plus k Superscript 6

(b) 1 Superscript k Baseline plus 2 Superscript k Baseline plus 3 Superscript k Baseline plus 4 Superscript k Baseline plus 5 Superscript k Baseline plus 6 Superscript k

(c) 2 Superscript k Baseline plus 3 Superscript k Baseline plus 4 Superscript k Baseline plus 5 Superscript k Baseline plus 6 Superscript k Baseline plus 7 Superscript k

(d) 2 plus 2 plus 2 plus 2 plus 2 plus 2 equals 12

(e) k Superscript 1 Baseline times k squared times k cubed times k Superscript 4 Baseline times k Superscript 5 Baseline times k Superscript 6 Baseline equals k Superscript 1 plus 2 plus 3 plus 4 plus 5 plus 6 Baseline equals k Superscript 21

(f) 2 times 2 times 2 times 2 times 2 times 2 equals 2 Superscript 6

Task 2

By writing out the terms explicitly show that

sigma summation Underscript i equals 1 Overscript i equals 5 Endscripts 3 i equals 3 sigma summation Underscript i equals 1 Overscript i equals 5 Endscripts i period

Show answer

sigma summation Underscript i equals 1 Overscript i equals 5 Endscripts 3 i equals 3 plus 6 plus 9 plus 12 plus 15 equals 3 times 1 plus 3 times 2 plus 3 times 3 plus 3 times 4 plus 3 times 5 equals 3 times left parenthesis 1 plus 2 plus 3 plus 4 plus 5 right parenthesis equals 3 sigma summation Underscript i equals 1 Overscript i equals 5 Endscripts i

Task 3

Write out fully, the following expressions:

(a) 3 m Superscript 4

(b) left parenthesis 3 m right parenthesis Superscript 4

Show answer

VideoVideo model answersDuration0:42

(a) 3 m Superscript 4

(b) 81 m Superscript 4

Task 4

Simplify the following expressions:

(a) b Superscript 5 Baseline times b squared times b

(b) b Superscript 5 Baseline times b squared times StartFraction b Over b cubed EndFraction

Show answer

(a) b Superscript 8

(b) b Superscript 5

Task 5

Remove the parentheses from the following expressions:

(a) left parenthesis 3 x right parenthesis squared

(b) left parenthesis 6 x y right parenthesis Superscript 4

(c) left parenthesis x cubed y Superscript 5 Baseline right parenthesis cubed

Show answer

(a) 9 x squared

(b) 6 Superscript 4 Baseline x Superscript 4 Baseline y Superscript 4

(c) x Superscript 9 Baseline y Superscript 15

Task 6

Show that left parenthesis minus x y right parenthesis cubed is equal to minus x cubed y cubed.

Show answer

left parenthesis minus x y right parenthesis cubed equals left parenthesis negative 1 right parenthesis cubed x cubed y cubed equals minus x cubed y cubed period

Task 7

Write each of the following expressions using a positive index:

(a) 2 Superscript negative 3

(b) StartFraction 1 Over 4 Superscript negative 3 Baseline EndFraction

(c) x Superscript negative 5

Show answer

VideoVideo model answersDuration0:42

(a) StartFraction 1 Over 2 cubed EndFraction

(b) 4 cubed

(c) StartFraction 1 Over x Superscript 5 Baseline EndFraction

Task 8

Simplify the following expressions:

(a) StartFraction a Superscript 8 Baseline a cubed Over a Superscript 5 Baseline EndFraction

(b) StartFraction a Superscript 8 Baseline b squared a cubed b Superscript 4 Baseline Over b Superscript 7 Baseline a Superscript 5 Baseline EndFraction

Show answer

(a) a Superscript 6

(b) StartFraction a Superscript 6 Baseline Over b EndFraction

Task 9

Evaluate the following:

(a) 144 Superscript 1 divided by 2

(b) 125 Superscript 1 divided by 3

Show answer

(a) 12

(b) 5

Task 10

Simplify the following:

(a) StartFraction StartRoot x EndRoot Over x cubed x squared EndFraction

(b) StartFraction x squared Over x Superscript negative 1 divided by 2 Baseline RootIndex 3 StartRoot x squared EndRoot EndFraction

Show answer

VideoVideo model answersDuration1:29

(a) StartFraction 1 Over x Superscript nine halves Baseline EndFraction equals StartFraction 1 Over StartRoot x Superscript 9 Baseline EndRoot EndFraction

(b) x Superscript 11 divided by 6

Sets

Introduction to sets

  • A set upper S is a well-defined, unordered collection of objects. We typically use curly brackets to denote sets, for example upper S equals StartSet 1 comma 2 EndSet.
  • The objects that make up the set are also known as elements of the set.
  • If x is an element of upper S, we can say that x belongs to upper S and write x element of upper S (the symbol element of reads "belongs to" or "in"). If, on the other hand, an element z does not belong to upper S we can write z not an element of upper S. To give an example, for upper S equals StartSet 1 comma 2 EndSet, 1 element of upper S, but 3 not an element of upper S.
  • A set may contain finitely many or infinitely many elements.
  • A set with no elements is called the empty set and is denoted by the symbol normal empty set.
  • The number of elements within a set upper S is called the cardinality of the set and is denoted by bold c a r d left parenthesis upper S right parenthesis or StartAbsoluteValue upper S EndAbsoluteValue.
  • Given sets upper S and upper T , we say that upper S is a subset of upper T if every element of upper S is also an element of upper T. We can then write upper S subset of upper T. In that case, we can also say that upper T is a superset of upper S; and write it as upper T superset of upper S. The diagram below (which is known as a Venn diagram) illustrates the definition.
Figure 1

Subset

  • Given sets upper S and upper T, their union upper S union upper T is the set of elements that are either in upper S or upper T (or in both).
Figure 2
  • Given sets upper S and upper T, their intersection upper S intersection upper T is the set of elements that are both in upper S and upper T.
Figure 3
  • A set upper S is called the complement of upper T if it contains all the elements that do not belong to it. The complement of upper T is written as upper T Superscript complement (or upper T overbar or upper T prime).
Figure 4
  • Given two sets, upper S and upper T, the difference upper S minus upper T contains all elements of upper S that are not contained in upper T. The set difference can be, more formally defined as the intersection of upper S and the complement of upper T, upper S minus upper T equals upper S intersection upper T Superscript complement.
Figure 5
Example 1

Let's assume that I asked 43 people if they like dogs or cats. 23 of them said they like dogs, 14 of them told me they like cats while there were 6 people who like both dogs and cats.

If we denote as upper C and upper D the sets referring to the people who like cats and dogs respectively; then we are given the following information: StartAbsoluteValue upper D EndAbsoluteValue equals 23, StartAbsoluteValue upper C EndAbsoluteValue equals 14 and StartAbsoluteValue upper D intersection upper C EndAbsoluteValue equals 6. Also, there are are 43 people in total. This information is shown on the diagram below (which is known as a Venn diagram).