Solveing a Venn Daigram When You Only Know a B C and the Unervisal

The Improving Mathematics Education in Schools (TIMES) Project

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Sets and Venn Diagrams

Number and Algebra : Module iYears : vii-8

June 2011

PDF Version of module

Assumed noesis

• Addition and subtraction of whole numbers.
• Familiarity with the English words

'and', 'or', 'not', 'all', 'if…and then'.

Motivation

In all sorts of situations we classify objects into sets of similar objects and count them. This procedure is the most bones motivation for learning the whole numbers and learning how to add and subtract them.

Such counting speedily throws upwards situations that may at offset seem contradictory.

'Concluding June, at that place were 15 windy days and twenty rainy days, yet v days were neither windy nor rainy.'

How can this be, when June merely has 30 days? A Venn diagram, and the language of sets, easily sorts this out.

Permit W be the set of windy days,
and R be the set up of rainy days.
Let East be the ready of days in June.
Then W and R; together have size 25, so
the overlap between Westward and R is 10.; The Venn diagram opposite displays; the whole situation.

The purpose of this module is to introduce language for talking about sets, and some notation for setting out calculations, so that counting bug such equally this can exist sorted out. The Venn diagram makes the situation easy to visualise.

Content

Describing and naming sets

A prepare is just a collection of objects, only we need some new words and symbols and diagrams to exist able to talk sensibly virtually sets.

In our ordinary language, we try to brand sense of the globe nosotros live in by classifying collections of things. English has many words for such collections. For example, nosotros speak of 'a flock of birds', 'a herd of cattle', 'a swarm of bees' and 'a colony of ants'.

We do a similar affair in mathematics, and classify numbers, geometrical figures and other things into collections that we call sets. The objects in these sets are chosen the elements of the set.

Describing a set

A gear up tin can be described past list all of its elements. For example,

S = { one, 3, 5, vii, nine },

which we read as 'Southward is the set whose elements are 1, iii, 5, 7 and 9'. The five elements of the gear up are separated by commas, and the list is enclosed between curly brackets.

A set tin besides exist described past writing a clarification of its elements betwixt curly brackets. Thus the set S higher up can as well be written as

S = { odd whole numbers less than 10 },

which we read as 'South is the set of odd whole numbers less than x'.

A set up must be well defined. This ways that our description of the elements of a prepare is clear and unambiguous. For example, { tall people } is not a set, because people tend to disagree about what 'tall' ways. An example of a well-defined set is

T = { letters in the English alphabet }.

Equal sets

Two sets are called equal if they have exactly the same elements. Thus following the usual convention that 'y' is not a vowel,

{ vowels in the English alphabet } = { a, e, i, o, u }

On the other hand, the sets { i, 3, 5 } and { i, 2, 3 } are not equal, because they have different elements. This is written equally

{ 1, three, v } ≠ { 1, 2, iii }.

The society in which the elements are written between the curly brackets does not matter at all. For example,

{ 1, 3, v, 7, 9 } = { three, 9, 7, five, 1 } = { v, ix, 1, 3, 7 }.

If an chemical element is listed more than in one case, it is but counted in one case. For example,

{ a, a, b } = { a, b }.

The set { a, a, b } has only the two elements a and b. The second mention of a is an unnecessary repetition and tin be ignored. It is normally considered poor annotation to list an element more than than once.

The symbols ∈ and ∉

The phrases 'is an element of' and 'is not an element of' occur so often in discussing sets that the special symbols ∈ and ∉ are used for them. For example, if A = { 3, iv, five, half dozen }, and so

3 ∈ A (Read this as '3 is an element of the fix A'.)

8 ∉ A (Read this as '8 is not an element of the set A'.)

Describing and naming sets

• A set is a collection of objects, called the elements of the set.
• A set must exist well defined, pregnant that its elements can be described and
  listed without ambiguity. For example:

{ 1, iii, five } and { letters of the English language alphabet }.

• Two sets are chosen equal if they accept exactly the same elements.
• The club is irrelevant.
• Whatever repetition of an chemical element is ignored.
• If a is an element of a ready S, we write a ∈ S.
• If b is not an element of a gear up S, we write b ∉ S.

EXERCISE one

a

Specify the fix A by listing its elements, where
A = { whole numbers less than 100 divisible past sixteen }.

b

Specify the fix B by giving a written description of its elements, where
B = { 0, 1, iv, nine, 16, 25 }.

c

Does the following judgement specify a set?
C = { whole numbers close to 50 }.

Finite and infinite sets

All the sets we have seen so far have been finite sets, meaning that we can list all their elements. Here are two more examples:

{ whole numbers between 2000 and 2005 } = { 2001, 2002, 2003, 2004 }

{ whole numbers between 2000 and 3000 } = { 2001, 2002, 2003,…, 2999 }

The three dots '…' in the 2nd example stand for the other 995 numbers in the set. We could have listed them all, but to save infinite we have used dots instead. This notation can only be used if it is completely clear what it ways, every bit in this situation.

A set tin can too be infinite − all that matters is that it is well divers. Here are ii examples of infinite sets:

{ even whole numbers } = { 0, 2, 4, half dozen, 8, x, …}

{ whole numbers greater than 2000 } = { 2001, 2002, 2003, 2004, …}

Both these sets are infinite because no matter how many elements we list, there are always more elements in the set that are non on our list. This time the dots '…' have a slightly different meaning, because they represent infinitely many elements that we could not peradventure list, no matter how long we tried.

The numbers of elements of a set

If S is a finite set, the symbol | S | stands for the number of elements of South. For example:

If S = { i, 3, v, 7, 9 }, then | S | = 5.

If A = { 1001, 1002, 1003, …, 3000 }, and so | A | = 2000.

If T = { letters in the English alphabet }, and so | T | = 26.

The set S = { 5 } is a one-chemical element set considering | Southward | = 1. It is important to distinguish between the number 5 and the set S = { 5 }:

5 ∈ S simply 5 ≠ Southward .

The empty set

The symbol ∅ represents the empty gear up, which is the gear up that has no elements at all. Nothing in the whole universe is an element of ∅:

| ∅ | = 0 and x ∉ ∅, no matter what x may be.

There is but one empty set up, considering any ii empty sets have exactly the same elements, so they must be equal to i some other.

Finite and Infinite sets

• A set up is called finite if we can list all of its elements.
• An space set has the property that no affair how many elements we list,
  there are always more elements in the ready that are not on our list.
• If S is a finite set, the symbol | South | stands for the number of elements of S.
• The ready with no elements is called the empty set, and is written as ∅.
  Thus | ∅ | = 0.
• A i-element set is a ready such equally Due south = { v } with | S | = 1.

EXERCISE 2

a

Use dots to help listing each set, and state whether it is finite or infinite.

i

B = { even numbers betwixt x 000 and 20 000 }

ii

A = { whole numbers that are multiples of 3 }

b

If the set South in each role is finite, write down | S |.

i

S = { primes }

ii

Due south = { even primes }

3

S = { fifty-fifty primes greater than 5 }

iv

S = { whole numbers less than 100 }

c

Let F be the set of fractions in simplest course between 0 and 1 that can be written with a single-digit denominator. Find F and | F |.

Subsets and Venn diagrams

Subsets of a gear up

Sets of things are often further subdivided. For case, owls are a particular type of bird, and then every owl is also a bird. Nosotros express this in the language of sets by maxim that the prepare of owls is a subset of the fix of birds.

A set S is called a subset of another set T if every element of S is an element of T. This is written equally

Southward ⊆ T (Read this every bit 'Southward is a subset of T'.)

The new symbol ⊆ means 'is a subset of'. Thus { owls } ⊆ { birds } because every owl is a bird. Similarly,

if A = { 2, iv, 6 } and B = { 0, 1, 2, three, 4, v, 6 }, then A ⊆ B,

because every chemical element of A is an element of B.

The judgement 'Due south is not a subset of T' is written as

Southward T.

This means that at least one element of South is not an element of T. For example,

{ birds } { flight creatures }

because an ostrich is a bird, but it does not fly. Similarly,

if A = { 0, 1, 2, three, 4 } and B = { 2, 3, iv, 5, half dozen }, then A B,

because 0 ∈ A, but 0 ∉ B.

The set itself and the empty prepare are always subsets

Any gear up S is a subset of itself, because every element of S is an element of S. For example:

{ birds } ⊆ { birds } and { i, two, three, 4, 5, vi } = { 1, 2, iii, 4, 5, 6 }.

Furthermore, the empty set ∅ is a subset of every set up S, because every element of the empty fix is an chemical element of South, in that location being no elements in ∅ at all. For example:

∅ ⊆ { birds } and ∅ ⊆ { one, two, 3, four, five, 6 }.

Every chemical element of the empty set is a bird, and every element of the empty set is one of the numbers 1, ii, 3, 4, 5 or 6.

Subsets and the words 'all' and 'if … then'

A argument about subsets can exist rewritten equally a sentence using the word 'all'.
For example,

{ owls } ⊆ { birds }		means		'All owls are birds.'
{ multiples of 4 } ⊆ { even numbers }		means		'All multiples of 4 are even.'
{ rectangles } ⊆ { rhombuses }		means		'Not all rectangles are rhombuses.'

They can likewise exist rewritten using the words 'if … and so'. For case,

{ owls } ⊆ { birds }		means		'If a brute is an owl, then it is a bird.'
{ multiples of 4 } ⊆ { even numbers }		means		'If a number is a multiple of iv, then information technology is even':
{ rectangles } ⊆ { rhombuses }		ways		'If a figure is a rectangle, then information technology may not exist a square.'

Venn diagrams

Diagrams make mathematics easier because they help united states of america to see the whole situation at a glance. The English mathematician John Venn (1834−1923) began using diagrams to stand for sets. His diagrams are now called Venn diagrams.

In nigh problems involving sets, it is convenient to choose a larger fix that contains all of the elements in all of the sets being considered. This larger set is called the universal set, and is usually given the symbol E. In a Venn diagram, the universal set up is generally fatigued as a large rectangle, and so other sets are represented by circles within this rectangle.

For example, if Five = { vowels }, nosotros could choose the universal set as E = { letters of the alphabet } and all the letters of the alphabet would and then need to be placed somewhere inside the rectangle, as shown below.

In the Venn diagram below, the universal set is Eastward = { 0, 1, 2, 3, four, 5, 6, 7, eight, ix, 10 }, and each of these numbers has been placed somewhere inside the rectangle.

The region inside the circle represents the ready A of odd whole numbers between 0 and ten. Thus we identify the numbers 1, three, 5, 7 and ix inside the circumvolve, because A = { 1, 3, 5, 7, 9 }. Outside the circle nosotros identify the other numbers 0, two, 4, six, viii and 10 that are in E just non in A.

Representing subsets on a Venn diagram

When we know that S is a subset of T, nosotros place the circle representing South inside the circle representing T. For example, let Southward = { 0, 1, 2 }, and T = { 0, i, 2, three, 4 }. So Southward is a subset of T, equally illustrated in the Venn diagram beneath.

Make certain that five, half dozen, seven, viii, 9 and x are placed exterior both circles>

Subsets and the number line

The whole numbers are the numbers 0, one, 2, 3,… These are frequently called the 'counting numbers', because they are the numbers we use when counting things. In detail, we accept been using these numbers to count the number of elements of finite sets. The number zero is the number of elements of the empty set.

The set of all whole numbers can exist represented by dots on the number line.

Any finite subset of set of whole numbers can be represented on the number line. For example, here is the prepare { 0, 1, 4 }.

Subsets of a st

• If all the elements of a set South are elements of some other set T, then Southward is called a subset of T. This is written every bit Southward ⊆ T.
• If at to the lowest degree one chemical element of South is not an element of T, then South is non a subset of T. This is written every bit South T.
• If South is any set up, then ∅ ⊆ Southward and Southward ⊆ S.
• A statement nearly a subset can exist rewritten using the words 'all' or 'if … so'.
• Subsets can exist represented using a Venn diagram.
• The set up { 0, 1, 2, iii, 4, … } of whole numbers is infinite.
• The set of whole numbers, and whatsoever finite subset of them, can be represented on the number line.

Do 3

a

Rewrite in gear up note:

i

All squares are rectangles.

two

Not all rectangles are rhombuses.

b

Rewrite in an English sentence using the words 'all' or 'non all':

i

{ whole number multiples of six } ⊆ { fifty-fifty whole numbers }.

2

{ square whole numbers } ⊆ { even whole numbers }.

c

Rewrite the statements in part (b) in an English sentence using the words 'if …, and so'.

d

Given the sets A = { 0, one, iv, 5 } and B = { 1, 4 }:

i

Draw a Venn diagram of A and B using the universal ready U = { 0, 1, 2, … , 8 }.

two

Graph A on the number line.

Complements, intersections and unions

The complement of a set

Suppose that a suitable universal set Due east has been called. The complement of a fix S
is the set of all elements of E that are not in Due south. The complement of S is written as S c.
For example,

If Due east = { messages } and V = { vowels }, and then V c = { consonants }

If E = { whole numbers } and O = { odd whole numbers },
then O c = {even whole numbers}.

Complement and the word 'not'

The give-and-take 'non' corresponds to the complement of a set. For example, in the 2 examples above,

5 c = { messages that are not vowels } = { consonants }

O c = { whole numbers that are non odd } = { even whole numbers }

The set Five c in the first example tin be represented on a Venn diagram as follows.

The intersection of 2 sets

The intersection of two sets A and B consists of all elements belonging to A and to B.
This is written as A ∩ B. For example, some musicians are singers and some play an instrument.

	If		A = { singers } and B = { instrumentalists }, so
			A ∩ B = { singers who play an instrument }.

Hither is an example using letters.

	If		V = { vowels } and F = { messages in 'dingo' }, then
			V ∪ F = { i, o }.

This concluding example tin be represented on a Venn diagram as follows.

Intersection and the word 'and'

The word 'and' tells united states that there is an intersection of 2 sets. For example:

{ singers } ∩ { instrumentalists } = { people who sing and play an instrument }

{ vowels } ∩ { letters of 'dingo' } = { messages that are vowels and are in 'dingo' }

The union of two sets

The union of two sets A and B consists of all elements belonging to A or to B. This is written every bit A ∪ B. Elements belonging to both set belong to the union. Continuing with the example of singers and instrumentalists:

If A = { singers } and B = { instrumentalists }, then A ∪ B = { musical performers }.

In the case of the sets of letters:

If V = { vowels } and F = { messages in 'dingo' }, then V &acup; F = { a, e, i, o, u, d, due north, g }.

Matrimony and the word 'or'

The give-and-take 'or' tells us that there is a matrimony of two sets. For example:

{ singers } ∪ { instrumentalists } = { people who sing or play an instrument }

{ vowels } ∪ { letters in 'dingo' } = { letters that are vowels or are in 'dingo' }

The word 'or' in mathematics always means 'and/or', so there is no need to add together 'or both' to these descriptions of the unions. For case,

	If		A = { 0, 2, 4, half dozen, viii, ten, 12, 14 } and B = { 0, 3, 6, nine, 12 }, so
			A ∪ B = { 0, two, 3, 4, 6, 8, nine, 10, 12, 14 }.

Here the elements half dozen and 12 are in both sets A and B.

Disjoint sets

Two sets are called disjoint if they accept no elements in common. For example:

The sets Due south = { 2, 4, 6, 8 } and T = { 1, iii, v, 7 } are disjoint.

Another way to define disjoint sets is to say that their intersection is the empty set,

Two sets A and B are disjoint if A ∩ B = ∅.

In the case higher up,

South ∩ T = ∅ because no number lies in both sets.

Complement, intersection and union

Let A and B be subsets of a suitable universal fix East.

• The complement A c is the set of all elements of Eastward that are not in A.
• The intersection A ∩ B is the set of all elements belonging to A and to B.
• The union A ∪ B is the set of all elements belonging to A or to B.
• In mathematics, the word 'or' ever means 'and/or', so all the elements that
  are in both sets are in the union.
• The sets A and B are called disjoint if they have no elements in mutual, that is,
  if A ∩ B = ∅.

Representing the complement on a Venn diagram

Allow A = { 1, three, 5, 7, 9 } be the set of odd whole numbers less than x, and take the universal gear up as E = { 0, one, 2, … , 10 }. Hither is the Venn diagram of the situation.

The region inside the circle represents the set A, and then we place the numbers 1, 3, 5, 7 and 9 inside the circle. Outside the circle, nosotros place the other numbers 0, 2, four, six, 8 and 10 that are non in A. Thus the region outside the circle represents the complement A c = {0, 2, 4, 6, 8, 10}.

Representing the intersection and union on a Venn diagram

The Venn diagram below shows the two sets

A = { 1, 3, 5, 7, 9 } and B = { 1, 2, three, 4, 5 }.

• The numbers one, iii and 5 lie in both sets, so nosotros identify them in the overlapping region of the two circles.
• The remaining numbers in A are 7 and nine. These are placed inside A, but outside B.
• The remaining numbers in B are two and 4. These are placed inside B, but outside A.

Thus the overlapping region represents the intersection A ∩ B = { one, 3, five }, and the two circles together stand for the marriage A ∪ B = { 1, 2, iii, 4, v, 7, 9 }.

The four remaining numbers 0, six, 8 and 10 are placed outside both circles.

Representing disjoint sets on a Venn diagram

When we know that two sets are disjoint, we stand for them by circles that exercise not intersect. For case, permit

P = { 0, i, 2, 3 } and Q = { 8, 9, 10 }

Then P and Q are disjoint, equally illustrated in the Venn diagram below.

Venn diagrams with complements, unions and intersections

• Sets are represented in a Venn diagram by circles drawn inside a rectangle representing the universal set.
• The region outside the circle represents the complement of the ready.
• The overlapping region of ii circles represents the intersection of the 2 sets.
• Ii circles together correspond the union of the two sets.
• When ii sets are disjoint, we can depict the two circles without any overlap.
• When one set is a subset of another, we can depict its circle within the circle of the other set.

EXERCISE 4

Let the universal fix exist E = {whole numbers less than 20 }, and let

A = { squares less than 20 }

B = { even numbers less than xx }

C = { odd squares less than xx }

a

Draw A and C on a Venn diagram, and place the numbers in the right regions.

b

Depict B and C on a Venn diagram, and place the numbers in the right regions.

c

Shade A ∩ B on a Venn diagram, and place the numbers in the right regions.

d

Shade A ∪ B on a Venn diagram, and place the numbers in the right regions.

Solving bug using a Venn diagram

Keeping count of elements of sets

Earlier solving issues with Venn diagrams, nosotros need to work out how to keep count of the elements of overlapping sets.

The upper diagram to the right shows two
sets A and B inside a universal set Due east, where

| A | = 6 and | B | = 7,

with 3 elements in the intersection A ∩ B.

The lower diagram to the right shows merely the
number of elements in each of the four regions.

These numbers are placed inside round brackets
so that they don't wait similar elements.

You tin see from the diagrams that

| A | = half-dozen and | B | = 7, simply | A ∪ B | ≠ 6 + 7.

The reason for this is that the elements inside the overlapping region A ∩ B should only be counted once, not twice. When we subtract the 3 elements of A ∩ B from the total, the calculation is then correct.

| A ∪ B | = half-dozen + 7 − 3 = 10.

EXAMPLE

In the diagram to the right,

| A | = 15, | B | = 25, | A ∩ B | = five and | E | = 50.

a

Insert the number of elements into each
of the four regions.

b

Hence find | A ∪ B | and | A ∩ B c |

Solution

a

Nosotros begin at the intersection and work outwards.

The intersection A ∩ B has 5 elements.

Hence the region of A outside A ∩ B has 10 elements,
and the region of B exterior A ∩ B has twenty elements.

This makes 35 elements so far, and then the outer region has 15 elements.

b

From the diagram, | A ∪ B | = 35 and | A ∩ B c | = 10.

EXERCISE 5

a

Depict a Venn diagram of 2 sets Southward and T

b

Given that | S | = 15, | T | = 20, | S ∪ T | = 25 and | Eastward | = 50, insert the number of elements into each of the four regions.

c

Hence observe | S ∩ T | and | S ∩ T c |.

Number of elements in the regions of a Venn diagram

•		The numbers of elements in the regions of a Venn diagram can exist washed by working systematically effectually the diagram.
•		The number of elements in the matrimony of two sets A and B is
•		Number of elements in A ∪ B = number of elements in A
•		Number of elements in A ∪ B	= number of elements in A
			+ number of elements in B
			− number of elements in A ∩ B.
•		Writing this formula in symbols, | A ∪ B | = | A | + | B | − | A ∩ B |.

Solving problems past drawing a Venn diagram

Many counting bug tin be solved by identifying the sets involved, then drawing upwards a Venn diagram to keep track of the numbers in the different regions of the diagram.

EXAMPLE

A travel agent surveyed 100 people to find out how many of them had visited the cities of
Melbourne and Brisbane. Thirty-one people had visited Melbourne, 26 people had been to Brisbane, and 12 people had visited both cities. Describe a Venn diagram to find the number of people who had visited:

a Melbourne or Brisbane

b Brisbane merely not Melbourne

c just one of the two cities

d neither city.

Solution

Let M be the set of people who had
visited Melbourne, and let B be the fix
of people who had visited Brisbane.
Let the universal set Eastward be the set of
people surveyed.

The information given in the question tin can now exist rewritten as

| M | = 31, | B | = 26, | K ∩ B | = 12 and | E | = 100.

Hence number in M just	= 31 − 12
	= 19
and number in B only	= 26 − 12
	= fourteen.

a Number visiting Melbourne or Brisbane = 19 + xiv +12 = 45.

b Number visiting Brisbane just = fourteen.

c Number visiting only ane urban center = nineteen + xiv = 33.

d Number visiting neither city = 100 − 45 = 55.

Problem solving using Venn diagrams

• First place the sets involved.
• Then construct a Venn diagram to go on runway of the numbers in the unlike regions of the diagram.

EXERCISE 6

Twenty-four people go on holidays. If 15 become swimming, 12 go fishing, and 6 practise neither, how many go pond and fishing? Describe a Venn diagram and fill in the number of people in all four regions.

Practice 7

In a certain school, in that location are 180 pupils in Year 7. One hundred and x pupils study French, 88 study German and 65 study Indonesian. 40 pupils study both French and German, 38 study German language and High german just. Observe the number of pupils who written report:

a		all three languages		b		Indonesian only
c		none of the languages		d		at least one language
due east		either ane ot two of the three languages.

Links Forrad

The examples in this module take shown how useful sets and Venn diagrams are in counting problems. Such problems volition continue to present themselves throughout secondary school.

The language of sets is also useful for understanding the relationships betwixt objects of different types. For example, we have met various sorts of numbers, and we can summarise some of our knowledge very concisely by writing

{ whole numbers } ⊆ { integers } ⊆ { rational numbers } ⊆ { existent numbers }.

The relationships amongst types of special quadrilaterals is more than complicated. Here are some statements about them.

{ squares } ⊆ { rectangles } ⊆ { parallelograms } ⊆ { trapezia }

{ rectangles } ∩ { rhombuses } = { squares }

If A = { convex kites } and B = { non-convex kites }, then

A ∩ B = ∅ and A ∪ B = { kites }

That is, the set of convex kites and the set of non-convex kites are disjoint, only their union is the fix of all kites.

Sets and probability

It is far easier to talk virtually probability using the linguistic communication of sets. The set of all outcomes is called the sample space, a subset of the sample space is chosen an result. Thus when nosotros throw iii coins, nosotros can accept the sample infinite as the fix

S = { HHH, HHT, HTH, HTT, THH, THT, TTH, TTT }

and the result 'throwing at least one head and at least one tail' is then the subset

East = { HHT, HTH, HTT, THH, THT, TTH }

Since each outcome is equally probable,

P(at least ane head and at to the lowest degree one tail) = = .

The event space of the complementary consequence 'throwing all heads or all tails' is the complement of the event infinite in the sample infinite, which we take equally the universal set, then

E c = { HHH, TTT }.

Since | E | + | E c | = | South |, information technology follows after dividing by | S | that P(East c) = one − P(East), and then

P(throwing all caput or all tails) = one − = .

Let F be the upshot 'throwing at least two heads'. So

F = { HHH, HHT, HTH, THH }

A Venn diagram is the all-time way to sort out the relationship between the 2 events East and F. We can then conclude that

P(Eastward and F) = 3 and P(E or F) = 7

Sets and Functions

When nosotros talk over a office, we ordinarily want to write down its domain − the prepare of all 10-values that we can substitute into it, and its range − the set up of all y-values that event from such substitutions.

For example, for the function y = x 2,

domain = { real numbers } and range = {y: y ≥ 0}.

The note used here for the range is 'ready-builder notation', which is no longer taught in school. Consequently we mostly avoid set notation altogether, and use instead less rigorous linguistic communication,

'The domain is all real numbers, and the range is y > 0.'

Speaking well-nigh the condition rather than about the set, however, tin can confuse some students, and it is frequently useful to demonstrate the set up theory ideas lying behind the abbreviated notation.

Sets and equations

Here are two inequalities involving absolute value and their solution.

| x | ≤ v (altitude from x to 0) ≤ 5		| x | ≥ 5 (distance from x to 0) ≥ five
10 ≥ −v and x ≤ 5.		10 ≤ −5 and x ≥ five.

If we use the language of solution sets, and pay attention to 'and' and 'or', we see that the solution of the first inequality is the intersection of two sets, and the solution of the second inequality is the matrimony of two sets. In set-builder notation, the solutions to the ii inequalities are

{ x: x ≥ −five } ∩ { x: x ≤ five } = { 10: −5 ≤ x ≤ v}, and

{ x: 10 ≤ −five } ∩ { 10: 10 ≥ 5 } = { ten: x £ −5 or x ≥ 5}.

At school, notwithstanding, nosotros simply write the solutions to the two inequalities as the weather condition alone,

−5 ≤ x ≤ v and x ≤ −5 or x ≥ 5

There are many like situations where the more precise language of sets may
help to clarify the solutions of equations and inequalities when difficulties are raised during discussions.

History and applications

Counting problems go dorsum to ancient times. Questions almost 'infinity' were also keenly discussed past mathematicians in the ancient world. The idea of developing a 'theory of sets', all the same, only began with publications of the German language mathematician Georg Cantor in the 1870s, who was encouraged in his work past Karl Weierstrass and Richard Dedekind, two of the greatest mathematicians of all time.

Cantor's work involved the amazing insight that there are infinitely many different types of infinity. In the hierarchy of infinities that he discovered, the infinity of the whole numbers is the smallest blazon of infinity, and is the same every bit the infinity of the integers and of the rational numbers. He was able to bear witness, quite simply, that the infinity of the real numbers is very much larger, and that the infinity of functions is much larger over again. His work caused a sensation and some Cosmic theologians criticised his work as jeopardising 'God'due south exclusive claim to supreme infinity'.

Cantor's results nigh types of infinity are spectacular and not particularly difficult. The topic is quite suitable every bit extension work at school, and the bones ideas take been presented in some details in Appendix 2 of the Module The Real Numbers.

Cantor's original version of set theory is now regarded as 'naive set theory', and contains contradictions. The nearly famous of these contradictions is called 'Russell's paradox', after the British philosopher and mathematician Bertrand Russell. It is a version of the ancient barber-paradox,

'A barber shaves all those who do not shave themselves. Who shaves the barber?'

and information technology works similar this:

'Sets that are members of themselves are rather unwelcome objects.

In order to distinguish such tricky sets from the ordinary, well-behaved sets,

allow S be the prepare of all sets that are not members of themselves.

Only when we consider the set S itself, we accept a problem.

If S is a fellow member of S, then S is not a fellow member of South.

If S is not a member of S, then S is a fellow member of Due south.

This is a contradiction.'

The best-known response, only past no ways the only response, to this problem and to the other difficulties of 'naive fix theory' is an alternative, extremely sophisticated, formulation of set theory called 'Zermelo-Fraenkel set theory', but it is hardly the perfect solution. While no contradictions have been constitute,many disturbing theorems have been proven. Most famously, Kurt Goedel proved in 1931 that it is impossible to prove that Zermelo-Fraenkel prepare theory, and indeed whatsoever system of axioms within which the whole numbers can be constructed, does not contain a contradiction!

Nonetheless, set theory is now taken as the accented stone-bottom foundation of mathematics, and every other mathematical idea is defined in terms of set theory. Thus despite the paradoxes of set theory, all concepts in geometry, arithmetic, algebra and calculus − and every other branch of modern mathematics − are defined in terms of sets, and have their logical basis in ready theory.

Answers to Exercises

EXERCISE i

a A = { 0, 16, 32, 48, 64, 80, 96 }.

b The well-nigh obvious reply is B = { square numbers less than 30 }.

c No, because I don't know precisely enough what 'shut to' ways.

Practice ii

a		i		A = { 10 002, 10 004, … , 19 998 } is finite. ii B = { 0, iii, 6, … } is infinite.
b		i		This set is infinite.		ii		| S | = i.
		iii		| Due south | = 0.		iv		| S | = 100.
c		F = , , , , , , , , , , , , , , , , , , , , , , , , , , so | F | = 27.

Exercise 3

a		i		{ squares } ⊆ { rectangles }.		2		{ rectangles } ⊆ { rhombuses }.
b		i		All multiples of 6 are fifty-fifty.		ii		Not all squares are fifty-fifty.
c		i		If a whole number is a multiple of six, then it is fifty-fifty.
		ii		If a whole number is a foursquare, so it may not be fifty-fifty.

d		i
		ii

EXERCISE 4

EXERCISE v

The marriage S ∪ T has 25 elements, whereas S has fifteen elements and T has 20 elements, then the overlap S ∩ T has 10 elements.

Hence the region of S outside S ∩ T has v elements, and the region of T outside Southward ∩ T has 10 elements. Hence the outer region has 50 − 25 = 25 elements.

c From the diagram, | S ∩ T | = 10 and | Due south ∪ Tc | = 40.

Practise half dozen

Since only 18 people are involved in swimming or fishing and 15 + 12 = 27, in that location are 9 people who go pond and fishing.

EXERCISE 7

a 9 b ten c 12 d 168 eastward 159

The Improving Mathematics Education in Schools (TIMES) Project 2009-2011 was funded by the Australian Government Department of Education, Employment and Workplace Relations.

The views expressed here are those of the author and do not necessarily represent the views of the Australian Government Department of Didactics, Employment and Workplace Relations.

© The University of Melbourne on behalf of the International Centre of Excellence for Education in Mathematics (Water ice-EM), the education division of the Australian Mathematical Sciences Found (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
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Source: http://amsi.org.au/teacher_modules/Sets_and_venn_diagrams.html

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