assyst math lesson-1

8/9/2019 ASSYST Math Lesson-1

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Mathematics for the Science of Complex Systems

presented by

The Open University and The University of Warwick

28th

March 2010

Session 1. Set theory in the science of complex systems.

Prepared for the course team by Jeffrey Johnson

Contents

1.1 Getting going. ......................................................................................................... 1

1.2 Sets and elements.......................................................................................2

1.3 An introduction to mathematical notation ....................................................3

1.4 The basic concepts and notation of set theory ............................................4

1.5 Representing sets and set operations by diagrams.....................................6

1.7 Subscripts and indexed sets........................................................................7

1.8 Classes of sets and the power set...............................................................8

1.9 Product sets ................................................................................................8

1.10 Theorems and chains of reasoning ...........................................................9

1.11 Multiple choice questions ........................................................................11

1.12 Conclusion to Session 1 ..........................................................................14

Answers to the SAQs ......................................................................................15

CSS - ASSYST Course on Mathematics for the science of complex systems. Page 1 of 17


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1.1 Getting going.

Sets are fundamental in mathematics. They are the basic building blocks of just about everything.

Sets are intimately tied up with logic, and therefore their use has a great deal to offer in the

formulation of theories of complex systems. Fortunately, the basic ideas of set theory are very

simple and easy to understand, so lets get going.

1.2 Sets and elements

Any well-defined collection of objects forms a set. The objects are called elements ormembers of

the set. Examples of sets include:

the British monarchs since the Norman conquest

the even numbers

people who work for the Treasury

the planets

cells in a bodythe letters of the alphabet

the students studying this course

Members of a set may themselves be sets. Sets of sets are often called classes. For example, one

could define a set of sets of people classified by their jobs, ages, incomes, or health.

The notion of being well definedlies at the heart of mathematics and, I think, science. It will be

defined more precisely later in the course. For the moment let us say that a setXis well defined

when for any elementx we can decide whether it is a member ofXor not. If there exists such a

procedure for deciding membership, then we will say that the set is groundedoreffectively

decidable.

As we shall see, this definition needs refining for multi-valued and time-dependent logics. For

example, Prince William does not belong to the set of British Kings, but he may belong to it at

some future date. Most sets have dynamic membership, the exceptions above being the even

numbers and the letters of the alphabet. We can argue whether or not the set of planets is fixed.

Certainly the set ofknown planets has changed through time. Whether or not this belongs to a

fixedset of planets with members that we dont yet know about is a matter for speculation.

Indeed, we can hypothesise that there is a universal setof planets that contains all the planets that

we know and all the others yet to be discovered. Then the set of known planets is a subset of the

universal set of planets.

SAQs are self-assessed questions, intended to help you learn. The answers are given at the end of

the document.

SAQ 1

(a) Is the set of even numbers grounded?(b) Is the set of planets grounded?

(c) Is the set of poor people grounded?



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1.3 An introduction to mathematical notation

Mathematicians make extensive use of specialised notation not as a secret language to hide the

obvious, but because using a symbol like = is much less tedious than writing out the word equals

when used many time. Many people are surprised to learn that mathematical notation is a

relatively recent thing. For example, in the passage below, Fermat is using aeq because the

modern equals sign, =, had not yet been invented1:

It was not until the second half of the 16th century that developments occurred in the use of

symbolic notations that could really merit description as the beginnings of algebra. Franois Vite

(1540-1603) was among the first clearly to be aware of the significance of symbols that

generalise numbers or magnitudes. He introduced the use of a vowel for a quantity assumed to beunknown or undetermined and consonants for quantities that are assumed to be given. The idea of

given quantities being represented by a letter is the beginning of the role of modern variables as

placeholders for values. Thus Boyer remarks Here we find for the first time in algebra a clear-

cut distinction between the important concept of a parameter and the idea of unknown quantity2.

In 1636 Fermat (who has studied Vite) was writing equations like

A in E aeq Z pl.

The Latin in means times and the pl is an abbreviation for planus3. He showed that the

corresponding locus is a hyperbola (c.f.xy = c2).4 Vite and Fermat (at this time) still regarded

the magnitudes associated with letters in a geometrical way so that a product was an area etc 5.

Mathematical notation is created, and it can be more or less good for its purpose. In the

introduction to Wittgenstein's Tractatus Logico-Philosophicus, Bertrand Russel writes ... a good

notation has a subtlety and suggestiveness which make it seem, at times, like a live teacher.

Notational irregularities are often the first sign of philosophical errors, and a perfect notation

would be a substitute for thought.. Not only is mathematical notation created, but the people who

create often it invest a considerable amount of ego in it. Murray Gell-Man writes that a scientist

would rather use another person's toothbrush than another scientist's nomenclature.6

When you have finished this course you will be in a position to invent your own notation torepresent the complex systems that interest you. If your system has properties not possessed by

any other system you will have to invent new notation, possibly with the help of a mathematician.

You may do this anyway, and later discover a system that shares properties with yours developed

using a different notation. Then you too will have to decide on whether you will change

toothbrushes for the greater good of science, or if you will obscure the interdisciplinary

connections and doggedly stick to your way of doing it. Its a tough choice. Fortunately before

you face such dilemmas, a lot of mathematical notation is standard, and easy to learn.

SAQ 2

Translate the following formula into words: 17 X 49 + 4 2 (19 13) X (2 + 6) /4.

1 According to Wikipedia, (http://en.wikipedia.org/wiki/Robert_Recorde, referenced 14th August 2006), the

Welshman, Robert Recorde, introduced the equals sign, =, in 1557 .2 Boyer, C.,A history of mathematics, Wiley, 1968.3 planus means plane or surface.4 Fermat, P.,Isagoge ad locos planos et solidos, 1679. [written in 1636]5 Beynon, M, Russ, S, Variables in mathematics and computer science, in The mathematical revolution

inspired by computing, J. Johnson and M. Loomes (eds), Claredon Press (Oxford), 1991.6 Gell-Mann, M., Plectics: The study of simplicity and complexity,Europhysics News Vol. 33 No. 1,

2002, http://www.europhysicsnews.com/full/13/article5/article5.html (referenced 14th August 2006).

http://en.wikipedia.org/wiki/Robert_Recordehttp://www.europhysicsnews.com/full/13/article5/article5.htmlhttp://www.europhysicsnews.com/full/13/article5/article5.htmlhttp://en.wikipedia.org/wiki/Robert_Recorde


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1.4 The basic concepts and notation of set theory

Usually sets are denoted by capital letters such asA,B, C, ,X, Y,Z, and their elements are

denoted by lower case letters such as a, b, c, ,x,y,z.

There are two ways to specify sets. The first is called set definition by extension and involveslisting the elements of the set in braces, e.g.:

{a, e, i, o, u }

{rook, knight, bishop, king, queen, pawn}

{London, Paris, Berlin, Rome, Madrid, Budapest, }

The second way involves stating the properties which characterise the elements of the set. This is

called set definition by intension, for example

V = {x |x is a vowel }

C = {x |x is a chess piece }

E = {x |x is a European capital city}

We writexXto mean x is an element of the setX or, equivalently, x belongs toX. We

writexXto mean that x does not belong toX. For example, we can write

HelsinkiE

ManchesterE

Two sets are equal when they have exactly the same elements. Formally,A =B, if and only ifa

A implies aB, and bB implies b A.

IfA =B thenB =A.

In set theory, all the sets under consideration are assumed to be contained in some large fixed set

called the universal setoruniverse, denoted U.

Above we suggested a universal set of planets, which is a useful idea for those who want to

analyse planetary systems. A sociologist might consider a universal set of people, and an

economist might consider a universal set of goods in a particular market. A complex systems

scientist might use all these sets as sub-universes of a larger universe.

The set with no elements is called the empty set, ornull set, which is denoted by the symbol,,

which is like a zero with a line through it.

For example, the set {x |x is a pig andx can fly } = .

A set can be empty with respect to a particular universe. In the previous example,x was restricted

to the universe of pigs.

The empty set is unique, meaning there is only one empty set. IfA andB were both empty sets,

then they have exactly the same elements (none), soA =B.



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SAQ 3 Read the following in words:

(a) Letp {x |x is a chess piece } andp {x |x white }. Thenp {x |x is a black chess piece}

(b) Is it true that {x |x is a Dutch city andx has a cricket team } = ?

A setA is a subset of the set B if every element ofA is an element ofB. This is written asAB

or, equivalently,BA.

A is apropersubset ofB ifB contains at least one element not possessed byA. This is written as

AB or, equivalently,BA.

For example, the set of women is a proper subset of people. The set of economically active

people is a proper subset of the set of people in Britain.

SAQ 4 Read the following in words.

Is it true that ifAB andBA thenA =B?

The intersection of two setsA andB, writtenAB is the set of elements that belong toA and

belong toB:

AB = {x |xA andxB }

For example, the intersection between the set of people living in poverty and the set of people

with bad health is the set of people living in poverty with bad health.

The intersection of the set of over eighty years old, and the set of premier division footballer is (to

my knowledge) empty: which could be written eighty_year_olds premier_division_footballers

= .

When two sets have empty intersection, i.e. they have no elements in common, they are said to be

disjoint.

The union of two setsA andB, writtenAB is the set of elements that belong to either of them:

AB = {x |xA orxB }

For example, the union of the adults of people who are men and the set of adults who are womenis the set of people who are men or women.

Let U be a universal set, and letA be a subset ofU. The complementofA in U is defined to be the

setAc = {x |xU andxA }, the set of elements that belong to U but do not belong toA.

Intersection and union are said to operate on sets, and they are often called set operators orset

operations.



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SAQ 5. Read the following in words

(a)AA =A

(b) (AB) C= A (BC)

(c)Ac = {x |xU andxA }

1.5 Representing sets and set operations by diagrams

A B A A B B

A

(a) AB (b) AB (c) AB (d)AB =

Figure 1. Illustrating basic set relationships and operations using diagrams

Sets are commonly represented by circles and other enclosed areas. For example Figure 1(a)

showsA as a subset ofB. Figure 1(b) shows two setsA andB with their intersection, Figure 1(c)

shows the union ofA andB, while Figure 1(d) illustrates two disjoint sets.

Figures 2(a), (b) and (c) show the universal set as a rectangle with other sets in it. This enables

the complements of sets to be drawn as the shaded areas shown.

U

A A B

U U

(a)Ac (b) (AB)c (c) (AB) c

Figure 2. Representing sets, their intersection, unions and complements by diagrams

Diagrams like those shown in Figures 1 and 2 are called Venn diagrams after the logician JohnVenn (1834 1923). Charles Dodgson (Lewis Carol) is said to have introduced the notion ofrectangle for the universal set7. In fact the use of circles to represent sets began well before the

nineteenth century. Some authors refer toEuler circles, andEuler diagrams which differ from

Venn diagrams by allowing some of the sets to be disjoint8.

7http://www.lewiscarroll.org/religion/venn.html

8 http://en.wikipedia.org/wiki/Euler_diagram



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SAQ 6

(a) draw a diagram to illustrateAB.

(b) draw a diagram with setsA,B and Cand shade the set (AB) C

(c) draw a diagram setsA,B and Cand shade the set A (BC).

(d) What can you conclude from (b) and (c)

The difference of two setsA andB, writtenA B is defined to be the set whose elements belong

toA but notB, A B = {x |xA andxB }. This can be illustrated as shown in Figure 3(a).

The symmetric difference (Figure 2(b)) of two sets is the set of elements that belong toA orB but

not both. It is written as9AB, where is the Greek symbol capital delta.

A B A B

(a)A B (a)AB

Figure 3. Diagrams shows set difference and symmetric difference.

SAQ 7

(a) Draw a diagram to show that (A B) (B A) =AB

(b) Draw a diagram to show that (AB) C= (AC) (B C)

Euler and Venn diagrams are very useful for illustrating and exploring the intersections andunions of sets, but the pictures do not represent rigorous proofs.

1.7 Subscripts and Indexed Sets

In mathematics there are usually so many things that we run out of letters to represent them all.

This is one of the reasons that Greek letters are used in mathematics its because all the roman

letters have been used for something else. Even so, there are not enough characters, so another

trick is used, namely subscripting.

LetXbe a finite set with n elements. Then let the elements ofXbe numbered from 1 to n, and

write the ith numbered element be represented by the symbolxi. Thus we can write

X= {xi | i = 1, , n} = {x1,x2,x3, .,xn}

9A number of notations are used for symmetric difference: http://mathworld.wolfram.com/SymmetricDifference.html



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which allows the elementsX to be represented by just one character,x with a subscript, or small

number or symbol on its its lower right side.

Apart from saving characters this has other advantages because the subscripted can be indexed

using an index set.

Let I be a set of numbers, I = { 1, 2, 3, 4, 5, , n }. Then we can writeX= {xi | iI }. Iis saidto be an index setfor the elements ofX.

1.8 Classes of sets and the power set

In complex systems there are many heterogeneous sets. Sets can be arranged in classes, where a

class is a set of sets, and there can be classes of classes of sets. Classes of sets are often

represented using a script font, e.g. below C is a class of sets indexed by the set IC. In this

example I have added a subscript to the index set to link it explicitly to the class it indexes.

C= { Ci | i IC}

One of the most important classes of set is thepower setof aX, which is the set of all subsets of

X. The power set ofXis often denoted by P(X). If a set has n elements, it can be shown that its

power set has 2n members, so the power set ofXis sometimes represented by the symbol 2X.

SAQ 8

(a) What is the power set of the set { a, b, c, d }?

(b) How many elements are there in the power set of the set { a, b, c, d }?

(c) How many elements are there in the power of { 1, 2, 3, 4, 5}?

1.9 Product Sets

Given two setsA andB theirproductis the set of ordered pairs (a, b), i.e.

AB = { (a, b) | aA and bB }

As an example, ifR is the real numbers, R R is the set of ordered pairs of numbers, (x,y) that

can be used to represent the two-dimensional plane. Similarly, one can form the product of three

copies ofR as R R R to obtain triples of numbers that can be used to represent three-

dimensional space. This idea of representing geometric space by pairs or triples of numbers is dueto Descartes, and the product is often called the Cartesian product.

Although the space we live in seems to have three dimensions, theres nothing to stop us forming

a product such as R R R R to obtain a 4-dimensional space. This could very useful for,say, an economic systems where there are n variables. These could be represented by an n-tuple,

(x1,x2, ,xn) as points in an n-dimensional space.



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It is not necessary for all the sets to be the same in a product.

Given a class of sets C= { Ci | i {1, 2, 3, ,n} }, theirproductis the set ofn-tuples

{ (c1, c2, , cn) | c1C1, c2C2, , cnCn }

The product of an indexed class of sets, C, is writteni I Ci.C

SAQ 9

(a) What the product of the sets { a, b, c } and { 1, 2 }.

(b) What is the product of the set { a1, a2, a3} and {b1, b2} ?

(c) What is the product set of { ai | i I } and { bj |j J }?

1.10 Theorems and chains of reasoning

Generally in mathematics, things can be deduced from definitions, and we call such deductionstheorems. For example

Theorem 1.1

(i) for all sets,A,AA

(ii) for all setsA,B and C, ifA B andB CthenA C

(iii)A B andB A if and only10 ifA =B.

Proof

(i) By definition,AB if and only ifxA impliesxB.

ButxA impliesxA, so by definitionAA.

(ii) SupposeAB andBC. Then aA implies aB. But aB implies aC. Therefore

Then aA implies aCandA C.

(iii) Part 1:AB andBA impliesA =B.

SupposeA B andBA.

Then aA implies aB and every element inA is an element inB.

Similarly, bB implies bA and every element inB belongs toA.

ThusA =B.

Part 2:A =B implies AB andBA.

SupposeA =B. Then, by (i),AB,

IfA =B thenB =A,

SinceB =A, again by (i),BA.

10if and only if is used so often in mathematics that it has its own abbreviation as iff. It means the implication goes

both ways. Here it can be read as ifA =B thenA B andB A, and ifA B andB A if thenA =B.



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Discussion

Things likeAA in (i) seem so self-evident sometimes its hard to construct a proof. The proof

here involves showing that all sets satisfy the requirements of the definition.

The proof of (ii) depends on the transitivity of implies. In logic, ifp implies q and q implies r

thenp implies r. Herep is aA, q is aB, and ris aC. So we deduce ifaA then aCwhich is the condition in the definition forA C.

Part 1 of (iii) underlies a standard technique to show two sets are equal: first one shows thatA is a

subset ofB and then one shows thatB is a subset ofA.

Part 2 illustrates a typical sleight of hand in mathematical proofs. Where did IfA =B thenB =

A come from? Its stated on Page 4. Its obvious, but it has not been proved. Can you prove it?

Finally, as usual, the little back square above is used to make it clear that the formal part of the

theorem and its proof are finished.

SAQ 10. Show that for all setsA,B and C, ifA B andB CthenA C

Set theory has two beautiful equalities calledDe Morgans Laws after their discoverer, Augustus

DeMorgan (1806-1871):

(AB)c =A

cBc

(AB)c =AcBc

This first of these is illustrated in Figure 2 where the complements are shown shaded. The union

of the two complements on the right is anything that is dark in eitherAc orB

c, and this is the same

as the complement of the intersection ofA andB.

Sometimes when I look at this diagram, its obvious that (AB)c =A

cBc, but other times I just

cant see it (although Ive known De Morgans Laws for years). So, dont worry if the diagram is

not very illuminating for you.

B= A

(AB)c = Ac Bc

Figure 2. Illustration of De Morgans Law (AB)c =AcBcBut is De Morgans Law really true? Is it true for all sets, and not just these pictures? Lets try to

show that it is.

LetA andB be any sets in a universe U.



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By definitionAB = {x |xA andxB }, (1)

(1) implies thatx (AB) if and only ifxA orx B or both. (2)

By definition of complements,xA orx B in (2) if and only ifxAc orx B

c. (3)

(2) and (3) imply thatx (AB) if and only ifxAc orx Bc. (4)

By definition of complements, (AB)c = {x | xUandx (AB) }. (5)

Substituting (4) into (5) gives

(AB)c = {x | xUand (xA

c orx Bc) } (7)

But {x | xUand (xAc orx B

c) } is the setAcB

c. So (8)

(AB)c = {x | xUand (xA

c orx Bc) }=A

cBc (9)

So, (AB)c =A

cBc (10)

SAQ 11

Show (ABC)c

=AcB

cC

c?

1.11 Multiple Choice Questions

You have now completed all the formal material for this session. Answer each of the following

multiple choice questions, by selecting what you think is the right answer. If you cannot answer

the question circle the x.

Each multiple choice question has one answer. Tick your choices

Q1. the set { a, e, i, o, u } is defined

(a) explicitly

(b) extensionally

(c) by the listing convention

(d) intensionally

(e) independently

(f) internally

(g) elementally

(h) externally(x) dont know

Q2. Who is said to have invented the equals sign?

(a) Pascal

(b) Fermat

(c) Leonardo

(d) Vite



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(e) Boyer

(f) Recorde

(g) Russ

(h) Newton

(x) dont know

Q3. What is the correct way to read x AB (CD)

(a) x belongs to the intersection ofA andB which is a subset ofCminusD(b)x belongs to the union ofA andB which is a proper subset ofD minus C

(c)x is a subset ofAB which belongs to the symmetric difference ofCandD

(d)x doesnt belong to the union ofA andB which is a subset of the difference ofCminusD(e)x belongs to the intersection ofA andB which is a subset of the difference ofCminusD(f)x belongs to the union ofA andB which is a subset of the difference of set Cminus setD

(g)x and CminusD belong to the union ofA andB(h) The symmetric difference ofCandD containsA unionB which containsx(x) dont know

Q4. Given a setXin a universe U, the complementofXis

(a)Xc =X- U

(b)Xc = U - X

(c)Xc = Uc - X

(d)Xc =XU

(e)Xc =XU

(f)Xc =XU

(x) dont know

Q5 Which of the following is correct

(a)AB =Ac Bc

(b)AB =AcB

c

(c)AB = (A B) (B A)

(d)AB = (A B) (B A)

(e)AB = (A B) (B A)

(e)AB = (A B) + (B A)

(x) dont know

Q6. The power set of the set { a, b, c} is

(a)

(b) { {a}, {b}, {c} }(c) { {a, b}, {a, c}, {b, c} }

(d) { {a}, {b}, {c}, {a, b}, {a, c}, {b, c} }

(e) { {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

(f) { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

(g) 2{ {a}, {b}, {c} }

(h) P(X)

(x) dont know



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Q7 The product of the sets of {a, b, c} and { x, y, z } is

(a) { (a, x), (a, y), (a, z), (b, x), (b, y), (b, z), (c, x), (c, y), (c, z) }

(b) { (a, x), (b, y), (c, z) }

(c) { (a, b, c), (x, y, z) }

(d) { (a, b), (a, c), (b, c), (x, y), (x, z), (y, z) }(e) { (a x), (a y), (a z), (b x), (b y), (b z), (c x), (c y), (c z) }

(x) dont know

Q8 Which of the following is true

(a) ifAB andBCthenA = C

(b) ifAB andBCthenAC

(c) ifA =B andBCthenA = C

(d) ifAB andBCthenAC

(e) ifAB andB CthenAC

(x) dont know

Q9 Which of the following is correct?

(a) (AB)c =A

B

(b) (AB)c =AcBc

(c) (AB)c =Ac Bc

(d) (AB)c =A

c Bc

(e) (AB)c =A

cBc

(f) (AB)c =A B

(x) dont know

Q10 Which of the following is correct?

(a) (AB)c =AB

(b) (AB)c =A

c +Bc

(c) (AB)c =A

cB

c

(d) (AB)c =A

cB

c

(e) (AB)c =A

cBc

(f) (AB)c =A

B

We are working on an interactive web site for you to submit your answers for marking and feedback

Jeffrey Johnson, Spring 2010

You can send your answers to

[email protected]



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1.12 Conclusion to Session 1

This concludes your work in Session 1. During it you have seen the following

the definitions of elements, sets and classes

the set membership and subset relationships

the intersection and union relationships

the complement of a set in its universe

the use of Venn diagrams

the power set of a set

the product of two or more sets

This is a lot of material to cover in one session, and it has generated a lot of new notation.

Although all of this may be new to you, I hope you have grasped some of the basic ideas, and that

you feel you can read the notation and make sense of it.

To finish this Session, please go through your answers to the multiple-choice questions, and enterthem on the End of Session survey along with your other answers and email the survey to

[email protected]

(Answers previously sent to [email protected] also be processed)

mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Answers to the SAQs

SAQ 1

Your answers may differ to mine:

(a) A procedure for seeing if a number is even is to see if it its last digit is 0, 2, 4, 6, or 8, so the

set is grounded.

(b) There are well established procedures for recognising the known planets, so this set is

grounded.

(c) There are definitions for people being poor in terms of them being below given income

thresholds, and in this respect the set is grounded. As the term is used in many conversations it is

not grounded.

SAQ 2

17 X 49 + 4 2 (19 13) X (2 + 6) /4

translates as

seventeen times forty nine plus four divided by 2 is not equal to thirteen subtracted from nineteen

times one quarter of two added to six

The point of this SAQ is that you already know how to speak arithmetic because you know

what the symbols mean. In this session you will encounter new symbols, but translating them into

words is done in the same way.

SAQ 3

(a) Letp {x |x is a chess piece } andp {x |x white }. Thenp {x |x is a black chess piece}This read as Letp belong to the set of chess pieces and letp belong to the set of white elements.

Thenp does not belong to the set of black chess pieces.

(b) Is it true that {x |x is a Dutch city andx has a cricket team } = ?

This reads as Is it true that the set of Dutch cities with a cricket team is empty?.

SAQ 4

Is it true that ifAB andBA thenA =B translates as

Is it true that ifA is a proper subset ofB andB is a proper subset ofA thenA equalsB?.

Its not! Its not possible forA to be a proper subset ofB and forB to be a proper subset ofA.

SAQ 5

(a)AA =A reads asA intersectionA equalsA

(b) (AB) C= A (BC) reads asA intersectionB intersected with CequalsA intersected

with B intersection C.

(c)Ac = {x |xU andxA } reads asA complement is the set with members that belong to the

universal set but do not belong toA.



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SAQ 6

(a) draw a diagram to illustrateAB.

B

A

(b) draw a diagram with setsA,B and Cand shade the set (AB) C

(c) draw a diagram setsA,B and Cand shade the set A (BC).

C

A

B

C

A

B

(d) What can you conclude from (b) and (c)

From these two diagrams I can conclude that

(AB) C A (BC).

SAQ 7

(a) Draw a diagram to show that (A B) (B A) =AB

A B A B BA

=

(A B) (B A) AB

(b) Draw a diagram to show that (AB) C= (AC) (B C)

A

C

B A

C

commonto both

B

(AB) C = (AC) (B C)

A

C

B



17/17

CSS - ASSYST Course on Mathematics for the science of complex systems Page 17 of 17

SAQ 8

(a) What is the power set of the set { a, b, c, d }?

P({ a, b, c, d } ) = { , {a}, {b}, {c}, {d}, {a,b}, {a,c}, {a,d}, {b,c}, {b,d}, {c,d},{a, b, c},

{a, b, d}, {a, c, d}, {b, c, d}, {a, b, c, d} }

(b) How many elements are there in the power set of the set { a, b, c, d }?

Including the empty set there are 24 = 16 sets in P({ a, b, c, d } ).

(c) How many elements are there in the power sets of { 1, 2, 3, 4, 5}?

Including the empty set there are 25 = 32 sets in P( { 1, 2, 3, 4, 5})

SAQ 9

(a) What the product of the sets { a, b, c } and { 1, 2 }.

{ a, b, c } { 1, 2 } = { (a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2) }

(b) What is the product of the set { a1, a2, a3} and {b1, b2} ?

{ a1, a2, a3} {b1, b2} = {(a1, b1), (a1, b2), (a2, b1), (a2, b2), (a3, b1), (a3, b2) }

(c) What if the product of the set { ai | i I } and { bj |j J }?

{ ai | i I } { bj |j J } = { (ai, bj) | for all i I and allj J }

SAQ 10.

Show that for all setsA,B and C, ifAB andBC thenAC

LetxA. ThenxB becauseA B.xB requiresxCbecauseB C. ThusAC.

Now we have too prove thatAB to obtainAC.

SinceAB there is an element bB and bA. SinceBC bC. Thus there exists bC

and bA , so thatAC, andA is a proper subset ofC,AC.

SAQ 11

Show (ABC)c =A

cBcC

c ?

LetD =BC.

Then by de Morgans laws, (AD)c =A

cDc

AcD

c =Ac (BC.)

c

By de Morgans Laws

Ac (BC.)

c =Ac (B

cC

c) =AcB

cC

c

assyst math lesson-1

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