Drawing Venn Diagrams for Sets

The Improving Mathematics Education in Schools (TIMES) Projection

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Sets and Venn Diagrams

Number and Algebra : Module iYears : 7-eight

June 2011

PDF Version of module

Assumed knowledge

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

'and', 'or', 'non', 'all', 'if…then'.

Motivation

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

Such counting quickly throws upwards situations that may at get-go seem contradictory.

'Final June, in that location were 15 windy days and 20 rainy days, all the same 5 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.

Allow Due west exist the set up of windy days,
and R be the set of rainy days.
Let E be the set of days in June.
So W and R; together accept size 25, and then
the overlap between W and R is 10.; The Venn diagram opposite displays; the whole situation.

The purpose of this module is to innovate language for talking about sets, and some notation for setting out calculations, so that counting problems such as this can be sorted out. The Venn diagram makes the state of affairs easy to visualise.

Content

Describing and naming sets

A gear up is just a collection of objects, but we demand some new words and symbols and diagrams to exist able to talk sensibly about sets.

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

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

Describing a set

A ready can be described by listing all of its elements. For instance,

Due south = { 1, 3, 5, 7, 9 },

which we read every bit 'Southward is the set whose elements are one, 3, 5, 7 and 9'. The five elements of the fix are separated past commas, and the listing is enclosed between curly brackets.

A set tin also exist described by writing a clarification of its elements between curly brackets. Thus the prepare S higher up tin also be written as

S = { odd whole numbers less than x },

which we read every bit 'S is the ready of odd whole numbers less than 10'.

A gear up must be well defined. This means that our description of the elements of a ready is clear and unambiguous. For example, { alpine people } is not a set, because people tend to disagree about what 'alpine' means. An instance 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 paw, the sets { one, 3, 5 } and { ane, 2, iii } are not equal, because they take dissimilar elements. This is written as

{ one, 3, 5 } ≠ { i, ii, 3 }.

The order in which the elements are written betwixt the curly brackets does not affair at all. For instance,

{ 1, 3, 5, 7, 9 } = { iii, 9, 7, 5, one } = { 5, nine, 1, iii, 7 }.

If an chemical element is listed more once, information technology is but counted once. For example,

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

The set { a, a, b } has only the two elements a and b. The 2nd mention of a is an unnecessary repetition and tin be ignored. Information technology is normally considered poor notation to list an chemical element more than one time.

The symbols ∈ and ∉

The phrases 'is an element of' and 'is non an element of' occur so oftentimes in discussing sets that the special symbols ∈ and ∉ are used for them. For case, if A = { 3, 4, v, 6 }, so

3 ∈ A (Read this as '3 is an chemical element of the set up A'.)

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

Describing and naming sets

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

{ 1, iii, 5 } and { messages of the English alphabet }.

• 2 sets are called equal if they take exactly the same elements.
• The social club is irrelevant.
• Any repetition of an element is ignored.
• If a is an element of a prepare S, we write a ∈ S.
• If b is not an element of a fix S, nosotros write b ∉ South.

EXERCISE 1

a

Specify the set A by listing its elements, where
A = { whole numbers less than 100 divisible by 16 }.

b

Specify the gear up B by giving a written description of its elements, where
B = { 0, 1, 4, 9, xvi, 25 }.

c

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

Finite and space sets

All the sets we take seen so far take been finite sets, significant that we can listing 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 second example correspond the other 995 numbers in the set. We could have listed them all, but to save space we have used dots instead. This notation can only be used if it is completely clear what information technology ways, as in this situation.

A set can also be infinite − all that matters is that information technology is well defined. Here are two examples of infinite sets:

{ even whole numbers } = { 0, 2, 4, 6, 8, 10, …}

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

Both these sets are infinite because no thing how many elements we list, there are ever more elements in the set that are not on our list. This time the dots '…' have a slightly different significant, because they stand for infinitely many elements that nosotros could not possibly 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 Southward. For example:

If S = { 1, 3, v, 7, 9 }, and then | S | = 5.

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

If T = { messages in the English alphabet }, then | T | = 26.

The set up S = { 5 } is a one-element gear up because | S | = ane. It is important to distinguish betwixt the number 5 and the set S = { 5 }:

5 ∈ S but 5 ≠ S .

The empty set

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

| ∅ | = 0 and x ∉ ∅, no matter what 10 may exist.

There is simply i empty set, considering any two empty sets take exactly the same elements, and so they must be equal to one another.

Finite and Infinite sets

• A ready is chosen finite if we tin list all of its elements.
• An space set has the belongings that no affair how many elements we listing,
  there are always more elements in the set that are non on our list.
• If South is a finite prepare, the symbol | South | stands for the number of elements of South.
• The fix with no elements is chosen the empty set, and is written as ∅.
  Thus | ∅ | = 0.
• A one-element set is a gear up such equally S = { 5 } with | Southward | = ane.

Exercise 2

a

Utilise dots to help list each set, and state whether it is finite or infinite.

i

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

ii

A = { whole numbers that are multiples of 3 }

b

If the fix S in each part is finite, write down | Southward |.

i

Due south = { primes }

ii

S = { even primes }

iii

Due south = { even primes greater than 5 }

iv

S = { whole numbers less than 100 }

c

Let F be the fix of fractions in simplest form between 0 and 1 that can be written with a unmarried-digit denominator. Find F and | F |.

Subsets and Venn diagrams

Subsets of a set

Sets of things are often further subdivided. For example, owls are a particular blazon of bird, and then every owl is also a bird. We limited this in the language of sets by proverb that the set of owls is a subset of the set of birds.

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

S ⊆ T (Read this equally 'S 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 = { two, iv, half-dozen } and B = { 0, 1, 2, iii, 4, 5, half-dozen }, so A ⊆ B,

because every element of A is an element of B.

The sentence 'S is not a subset of T' is written as

S T.

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

{ birds } { flying creatures }

because an ostrich is a bird, but information technology does not wing. Similarly,

if A = { 0, one, 2, three, iv } and B = { two, 3, iv, five, half-dozen }, then A B,

because 0 ∈ A, but 0 ∉ B.

The set up itself and the empty set are always subsets

Any set S is a subset of itself, considering every chemical element of S is an element of S. For example:

{ birds } ⊆ { birds } and { 1, two, 3, four, 5, half-dozen } = { one, 2, 3, 4, v, 6 }.

Furthermore, the empty set up ∅ is a subset of every gear up Southward, because every element of the empty set is an chemical element of S, there beingness no elements in ∅ at all. For instance:

∅ ⊆ { birds } and ∅ ⊆ { 1, 2, 3, 4, v, 6 }.

Every element of the empty set is a bird, and every element of the empty prepare is 1 of the numbers i, 2, 3, 4, five or 6.

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

A statement nigh subsets tin be rewritten as a sentence using the word 'all'.
For example,

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

They tin also exist rewritten using the words 'if … then'. For example,

{ owls } ⊆ { birds }		ways		'If a creature is an owl, and so it is a bird.'
{ multiples of 4 } ⊆ { even numbers }		means		'If a number is a multiple of 4, and then it is even':
{ rectangles } ⊆ { rhombuses }		means		'If a figure is a rectangle, then it may not be a square.'

Venn diagrams

Diagrams make mathematics easier because they assist the states to see the whole state of affairs at a glance. The English language mathematician John Venn (1834−1923) began using diagrams to stand for sets. His diagrams are at present chosen Venn diagrams.

In most problems involving sets, it is convenient to choose a larger set that contains all of the elements in all of the sets existence 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 drawn as a large rectangle, and so other sets are represented past circles within this rectangle.

For instance, if V = { vowels }, we could choose the universal set as East = { letters of the alphabet } and all the letters of the alphabet would then need to be placed somewhere inside the rectangle, as shown below.

In the Venn diagram below, the universal set is E = { 0, one, 2, three, four, 5, 6, 7, 8, 9, 10 }, and each of these numbers has been placed somewhere within the rectangle.

The region within the circumvolve represents the prepare A of odd whole numbers betwixt 0 and 10. Thus we identify the numbers 1, 3, five, 7 and 9 inside the circle, because A = { i, 3, five, 7, nine }. Outside the circle nosotros place the other numbers 0, ii, 4, 6, 8 and 10 that are in East just not in A.

Representing subsets on a Venn diagram

When we know that S is a subset of T, nosotros place the circle representing S within the circle representing T. For example, permit S = { 0, 1, 2 }, and T = { 0, 1, 2, iii, 4 }. Then South is a subset of T, equally illustrated in the Venn diagram below.

Make sure that five, 6, vii, eight, ix and 10 are placed outside both circles>

Subsets and the number line

The whole numbers are the numbers 0, 1, two, iii,… These are often called the 'counting numbers', because they are the numbers we utilise when counting things. In detail, we have 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 tin be 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 set { 0, 1, 4 }.

Subsets of a st

• If all the elements of a set up S are elements of another set T, then S is called a subset of T. This is written every bit Southward ⊆ T.
• If at least 1 element of S is not an element of T, then Southward is not a subset of T. This is written as S T.
• If S is any gear up, then ∅ ⊆ Due south and S ⊆ S.
• A statement near a subset can be rewritten using the words 'all' or 'if … and so'.
• Subsets tin be represented using a Venn diagram.
• The ready { 0, one, 2, 3, 4, … } of whole numbers is infinite.
• The set of whole numbers, and any finite subset of them, can be represented on the number line.

Practice 3

a

Rewrite in set note:

i

All squares are rectangles.

two

Not all rectangles are rhombuses.

b

Rewrite in an English sentence using the words 'all' or 'not all':

i

{ whole number multiples of 6 } ⊆ { even whole numbers }.

ii

{ foursquare whole numbers } ⊆ { even whole numbers }.

c

Rewrite the statements in part (b) in an English language judgement using the words 'if …, then'.

d

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

i

Depict a Venn diagram of A and B using the universal set U = { 0, 1, 2, … , eight }.

ii

Graph A on the number line.

Complements, intersections and unions

The complement of a set

Suppose that a suitable universal fix Eastward has been called. The complement of a set S
is the set up of all elements of E that are non in S. The complement of Southward is written as Southward c.
For example,

If E = { messages } and Five = { vowels }, then Five c = { consonants }

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

Complement and the discussion 'not'

The word 'non' corresponds to the complement of a set. For example, in the 2 examples above,

Five c = { letters that are not vowels } = { consonants }

O c = { whole numbers that are not odd } = { even whole numbers }

The set V c in the start example can exist represented on a Venn diagram as follows.

The intersection of two sets

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

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

Here is an example using letters.

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

This last example can be represented on a Venn diagram as follows.

Intersection and the word 'and'

The word 'and' tells u.s. that at that place is an intersection of two sets. For case:

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

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

The marriage 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. Standing 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 messages:

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

Union and the word 'or'

The discussion 'or' tells us that in that location is a union of ii 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 discussion 'or' in mathematics always means 'and/or', so in that location is no need to add 'or both' to these descriptions of the unions. For example,

	If		A = { 0, two, 4, 6, 8, 10, 12, fourteen } and B = { 0, 3, 6, nine, 12 }, so
			A ∪ B = { 0, 2, 3, 4, vi, viii, 9, ten, 12, 14 }.

Here the elements 6 and 12 are in both sets A and B.

Disjoint sets

Two sets are called disjoint if they have no elements in mutual. For instance:

The sets S = { 2, 4, 6, viii } and T = { 1, 3, 5, 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 example above,

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

Complement, intersection and union

Let A and B be subsets of a suitable universal set E.

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

Representing the complement on a Venn diagram

Permit A = { 1, iii, v, 7, ix } be the set of odd whole numbers less than x, and take the universal fix as E = { 0, 1, 2, … , 10 }. Here is the Venn diagram of the situation.

The region inside the circumvolve represents the set A, so we place the numbers 1, three, 5, 7 and 9 inside the circle. Exterior the circle, we place the other numbers 0, 2, 4, vi, viii and 10 that are non in A. Thus the region exterior the circle represents the complement A c = {0, ii, 4, 6, eight, 10}.

Representing the intersection and union on a Venn diagram

The Venn diagram beneath shows the two sets

A = { ane, three, 5, 7, 9 } and B = { one, ii, 3, 4, 5 }.

• The numbers 1, 3 and 5 lie in both sets, and so nosotros place them in the overlapping region of the two circles.
• The remaining numbers in A are vii and 9. 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 = { i, three, 5 }, and the ii circles together represent the union A ∪ B = { one, 2, three, 4, 5, vii, 9 }.

The four remaining numbers 0, half dozen, 8 and 10 are placed exterior both circles.

Representing disjoint sets on a Venn diagram

When we know that two sets are disjoint, we represent them by circles that exercise not intersect. For example, let

P = { 0, one, 2, three } and Q = { 8, nine, ten }

Then P and Q are disjoint, as illustrated in the Venn diagram beneath.

Venn diagrams with complements, unions and intersections

• Sets are represented in a Venn diagram by circles drawn within a rectangle representing the universal fix.
• The region exterior the circle represents the complement of the set.
• The overlapping region of two circles represents the intersection of the two sets.
• 2 circles together represent the wedlock of the ii sets.
• When two sets are disjoint, we tin can draw the ii circles without whatsoever overlap.
• When one set is a subset of another, nosotros tin draw its circle inside the circle of the other set.

EXERCISE 4

Allow the universal set be E = {whole numbers less than 20 }, and let

A = { squares less than 20 }

B = { even numbers less than xx }

C = { odd squares less than 20 }

a

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

b

Describe 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 correct regions.

d

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

Solving issues using a Venn diagram

Keeping count of elements of sets

Before solving problems with Venn diagrams, we need to work out how to go along count of the elements of overlapping sets.

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

| A | = 6 and | B | = seven,

with 3 elements in the intersection A ∩ B.

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

These numbers are placed inside round brackets
and then that they don't expect similar elements.

You can see from the diagrams that

| A | = half-dozen and | B | = seven, but | 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 nosotros subtract the three elements of A ∩ B from the total, the adding is then correct.

| A ∪ B | = 6 + seven − 3 = 10.

Example

In the diagram to the right,

| A | = 15, | B | = 25, | A ∩ B | = 5 and | Due east | = l.

a

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

b

Hence find | A ∪ B | and | A ∩ B c |

Solution

a

Nosotros brainstorm at the intersection and work outwards.

The intersection A ∩ B has 5 elements.

Hence the region of A exterior A ∩ B has x elements,
and the region of B outside A ∩ B has 20 elements.

This makes 35 elements then far, then the outer region has fifteen elements.

b

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

Practise v

a

Draw a Venn diagram of two sets S and T

b

Given that | S | = xv, | T | = 20, | S ∪ T | = 25 and | East | = l, insert the number of elements into each of the four regions.

c

Hence notice | 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 tin can be done by working systematically around the diagram.
•		The number of elements in the union 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 exist solved past identifying the sets involved, and so drawing upward a Venn diagram to go on runway of the numbers in the different regions of the diagram.

EXAMPLE

A travel amanuensis 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. Draw a Venn diagram to find the number of people who had visited:

a Melbourne or Brisbane

b Brisbane merely non Melbourne

c only one of the ii cities

d neither city.

Solution

Permit M exist the set of people who had
visited Melbourne, and let B be the set
of people who had visited Brisbane.
Let the universal set E be the gear up of
people surveyed.

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

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

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

a Number visiting Melbourne or Brisbane = 19 + 14 +12 = 45.

b Number visiting Brisbane simply = xiv.

c Number visiting just one urban center = 19 + 14 = 33.

d Number visiting neither urban center = 100 − 45 = 55.

Problem solving using Venn diagrams

• Get-go identify the sets involved.
• Then construct a Venn diagram to keep runway of the numbers in the different regions of the diagram.

EXERCISE half-dozen

Twenty-four people keep holidays. If 15 become pond, 12 go angling, and half dozen do neither, how many go swimming and line-fishing? Draw a Venn diagram and fill up in the number of people in all four regions.

EXERCISE 7

In a certain school, at that place are 180 pupils in Year 7. One hundred and ten pupils study French, 88 report German and 65 study Indonesian. Forty pupils written report both French and High german, 38 written report German and High german simply. Find the number of pupils who study:

a		all three languages		b		Indonesian only
c		none of the languages		d		at to the lowest degree one linguistic communication
due east		either i ot two of the three languages.

Links Forward

The examples in this module have shown how useful sets and Venn diagrams are in counting problems. Such problems will go on to present themselves throughout secondary school.

The language of sets is besides useful for understanding the relationships between objects of unlike types. For instance, we accept met various sorts of numbers, and we can summarise some of our knowledge very concisely by writing

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

The relationships amidst types of special quadrilaterals is more complicated. Hither are some statements about them.

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

{ rectangles } ∩ { rhombuses } = { squares }

If A = { convex kites } and B = { not-convex kites }, then

A ∩ B = ∅ and A ∪ B = { kites }

That is, the ready of convex kites and the gear up of non-convex kites are disjoint, but their union is the gear up of all kites.

Sets and probability

It is far easier to talk well-nigh probability using the linguistic communication of sets. The set of all outcomes is called the sample space, a subset of the sample infinite is called an event. Thus when we throw three coins, we tin can take the sample space as the set

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

and the issue 'throwing at least one head and at least one tail' is and so the subset

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

Since each outcome is equally likely,

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

The consequence space of the complementary result 'throwing all heads or all tails' is the complement of the result space in the sample space, which we accept as the universal set, so

East c = { HHH, TTT }.

Since | Eastward | + | East c | = | Southward |, it follows subsequently dividing by | Southward | that P(E c) = 1 − P(E), so

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

Permit F be the issue 'throwing at least ii heads'. Then

F = { HHH, HHT, HTH, THH }

A Venn diagram is the best way to sort out the relationship between the ii events East and F. We tin can then conclude that

P(E and F) = three and P(E or F) = 7

Sets and Functions

When we discuss a function, we usually desire to write downward its domain − the prepare of all x-values that nosotros can substitute into it, and its range − the set of all y-values that issue from such substitutions.

For example, for the function y = 10 2,

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

The notation used here for the range is 'gear up-architect note', which is no longer taught in school. Consequently we mostly avert set annotation altogether, and use instead less rigorous language,

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

Speaking almost the status rather than almost the set, however, tin confuse some students, and it is often useful to demonstrate the set theory ideas lying backside the abbreviated notation.

Sets and equations

Here are two inequalities involving absolute value and their solution.

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

If nosotros employ the language of solution sets, and pay attention to 'and' and 'or', we run across that the solution of the starting time inequality is the intersection of 2 sets, and the solution of the 2nd inequality is the union of two sets. In set-architect notation, the solutions to the two inequalities are

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

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

At schoolhouse, still, we just write the solutions to the two inequalities every bit the weather solitary,

−five ≤ x ≤ 5 and x ≤ −5 or x ≥ five

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

History and applications

Counting issues go dorsum to aboriginal times. Questions about 'infinity' were likewise keenly discussed by mathematicians in the aboriginal earth. The idea of developing a 'theory of sets', even so, but began with publications of the German mathematician Georg Cantor in the 1870s, who was encouraged in his work by Karl Weierstrass and Richard Dedekind, two of the greatest mathematicians of all time.

Cantor's work involved the astonishing insight that there are infinitely many different types of infinity. In the bureaucracy of infinities that he discovered, the infinity of the whole numbers is the smallest type of infinity, and is the aforementioned as the infinity of the integers and of the rational numbers. He was able to evidence, quite simply, that the infinity of the existent numbers is very much larger, and that the infinity of functions is much larger again. His piece of work caused a sensation and some Cosmic theologians criticised his piece of work as jeopardising 'God'southward exclusive claim to supreme infinity'.

Cantor's results about types of infinity are spectacular and non particularly difficult. The topic is quite suitable equally extension piece of work at school, and the basic ideas have been presented in some details in Appendix two of the Module The Existent Numbers.

Cantor's original version of set theory is now regarded as 'naive gear up theory', and contains contradictions. The most famous of these contradictions is called 'Russell's paradox', subsequently 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 it works like 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 Southward be the set of all sets that are not members of themselves.

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

If S is a member of Southward, then S is not a member of S.

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

This is a contradiction.'

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

Nevertheless, prepare theory is now taken as the absolute rock-bottom foundation of mathematics, and every other mathematical idea is defined in terms of fix theory. Thus despite the paradoxes of set theory, all concepts in geometry, arithmetics, algebra and calculus − and every other co-operative of modernistic mathematics − are defined in terms of sets, and have their logical basis in set theory.

Answers to Exercises

EXERCISE 1

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

b The most obvious answer is B = { square numbers less than 30 }.

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

EXERCISE ii

a		i		A = { 10 002, ten 004, … , 19 998 } is finite. two B = { 0, three, 6, … } is infinite.
b		i		This set is infinite.		2		| Due south | = 1.
		three		| Southward | = 0.		iv		| South | = 100.
c		F = , , , , , , , , , , , , , , , , , , , , , , , , , , so | F | = 27.

Practise 3

a		i		{ squares } ⊆ { rectangles }.		2		{ rectangles } ⊆ { rhombuses }.
b		i		All multiples of 6 are fifty-fifty.		ii		Non all squares are even.
c		i		If a whole number is a multiple of 6, then information technology is even.
		ii		If a whole number is a square, then information technology may not be even.

d		i
		ii

Do 4

Practise 5

The union Southward ∪ T has 25 elements, whereas S has 15 elements and T has 20 elements, so the overlap S ∩ T has 10 elements.

Hence the region of S outside S ∩ T has five elements, and the region of T outside S ∩ T has x elements. Hence the outer region has 50 − 25 = 25 elements.

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

Exercise 6

Since only eighteen people are involved in swimming or fishing and xv + 12 = 27, there are nine people who go swimming and fishing.

Exercise 7

a 9 b 10 c 12 d 168 e 159

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

The views expressed here are those of the writer and do not necessarily stand for the views of the Australian Government Department of Education, Employment and Workplace Relations.

© The University of Melbourne on behalf of the International Centre of Excellence for Instruction in Mathematics (ICE-EM), the education division of the Australian Mathematical Sciences Constitute (AMSI), 2010 (except where otherwise indicated). This work is licensed under the Artistic Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
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Drawing Venn Diagrams for Sets

Source: http://amsi.org.au/teacher_modules/Sets_and_venn_diagrams.html