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WOODBROOK SECONDARY SCHOOL MATHEMATICS SET THEORY FORM 4 1 | Page Definition of a Set A set is a collection of well defined objects. Capital letters are used to denote a set and set brackets {} are used to list the objects that can be found in that set. Building a Set A set must be defined. Ex. Let A be the set of the first 5 even numbers. Next find the objects that belong to this set. Ex. 2, 4, 6 ,8 ,10 State the set. Ex. A = {2, 4, 6, 8, 10} Common Sets The set of natural numbers or counting numbers: N = {1, 2, 3, 4, 5, …..} The set of whole numbers: W = {0, 1, 2, 3, 4, 5, …..} The set of integers: Z = {….., -3, -2, -1, 0, 1, 2, 3,…..} 1. Let M be the set of the first 7 prime numbers. Using set notation, state the set M. M = { } Even numbers are numbers that can be divided by 2 without leaving a remainder. There are 5 objects in ascending order in this set and all the objects are divisible by 2 A prime number is a number that can be divided by 1 and itself.

WOODBROOK SECONDARY SCHOOL MATHEMATICS SET THEORY FORM 4 · WOODBROOK SECONDARY SCHOOL MATHEMATICS SET THEORY FORM 4 1 | P a g e Definition of a Set A set is a collection of well

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WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

1 | P a g e

Definition of a Set

A set is a collection of well defined objects.

Capital letters are used to denote a set and set brackets {} are used to list the objects that can be

found in that set.

Building a Set

A set must be defined.

Ex. Let A be the set of the first 5 even numbers.

Next find the objects that belong to this set.

Ex. 2, 4, 6 ,8 ,10

State the set.

Ex. A = {2, 4, 6, 8, 10}

Common Sets

The set of natural numbers or counting numbers: N = {1, 2, 3, 4, 5, …..}

The set of whole numbers: W = {0, 1, 2, 3, 4, 5, …..}

The set of integers: Z = {….., -3, -2, -1, 0, 1, 2, 3,…..}

1. Let M be the set of the first 7 prime numbers. Using set notation, state the set M.

M = { }

Even numbers are

numbers that can be

divided by 2 without

leaving a remainder.

There are 5 objects in

ascending order in this

set and all the objects

are divisible by 2

A prime number is a

number that can be

divided by 1 and itself.

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

2 | P a g e

2. Let P be the set of 5 common flavours of ice-cream. Using set notation, state the set P.

P = { }

3. Let K be the set of 5 popular brands of cars in Trinidad. Using set notation, state the set

K.

K={ }

4. Let G be the set of the first 10 odd numbers.

Using set notation, state the set G.

G = { }

Belonging to a Set

An object that belongs to a set can be represented using the symbol ‘∈’ otherwise if it does not

belong to the set, the symbol ‘∉’ is used.

Let A be defined as follows: A = {2, 4, 6, 8, 10}

Determine which of the numbers from 1 through 10 belongs to the set A.

By inspection,

2 ∈A,

4 ∈A,

6 ∈A,

8 ∈A,

10 ∈A

1∉A

3 ∉A

5 ∉A

7 ∉A

9 ∉A

An odd number is a

number that leaves a

remainder of 1 when

divided by 2

These elements

can be found in A

These elements cannot

be found in A

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

3 | P a g e

The Empty or Null Set

Consider an empty school bag or an empty wallet or an empty room. If one was to look for a

book or money or a person respectively, nothing will be found.

The empty set, therefore, is the set that contains no objects and is given the set symbol, {∅}.

N.B. The empty set is a subset of all sets.

Let A be the set of pigs that fly. Using set notation, state the objects that belong to A.

Since there are no pigs that fly, A = {∅}.

Let D be the set of all dragons that could fly. Using set notation, state the objects that belong to

D.

Since there are no dragons that fly, D = {∅}.

Finite Set

A finite set is a set where there is a constriction on the number of objects that can be placed in a

set.

Ex. Let B be the set of the first 4 multiples of 5.

B = { }

The

constriction

here is the

“first 4”

Multiples are what you get

after multiplying a number

by a positive integer.

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

4 | P a g e

Infinite Set

An infinite set is a set that has no restrictions on what can be placed in a set.

Examples of infinite sets can be identified by the use of the three dots (...) and means to continue

on.

N = {1, 2, 3, 4, 5,…}

W = (0 , 1, 2, 3, 4, 5,…}

The Universal Set

The universal set, denoted by U, is the set that contains all objects or everything.

It is useful to note that the word ‘everything’ refers to what is relevant to the question.

All sets are taken from the universal set.

Ex. Let U be the set of natural numbers.

U = {1, 2, 3, 4, 5 , 6, 7, 8, 9, 10,…}

Sets can be built from the set U:

The set of even numbers between 1 and 20 – { }

The set of the first 10 prime numbers – { }

The set of odd numbers between 50 and 60 – { }

The set of the first 8 multiples of 5 – { }

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

5 | P a g e

Subsets

A subset of any set U is the set that contains all or some of the objects that can be found in U,

where U is the universal set.

Alternatively, Let A & B be two sets, A is a subset of B if all the objects in A can be found in B.

This is denoted by, A ⊆ 𝐵 otherwise if no elements in A can be found in B, then this is denoted

by A ⊈ 𝐵.

Proper Subsets

Let A & B be two sets, A is a proper subset of B if the objects in A can be found in B. However

not all the objects in B can be found in A.

This is denoted by, A ⊂ 𝐵 otherwise if no elements in B can be found in A, then this is denoted

by A ⊄ 𝐵.

The following example will illustrate.

Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}?

1 ∈ A, and 1 ∈ B as well.

3 ∈ A and 3 ∈ B.

4 ∈ A, and 4 ∈ B.

That's all the elements of A, and every single one is in B.

Yes, A is a subset of B ie 𝐴 ⊂ 𝐵

NB. Although 2 ∈ B, it cannot be found in A, ie 2∉ 𝐴

To determine the number of subsets that can be formed from any given set, the following

formula is useful:

2𝑛

n represents the

number of elements in

the set.

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

6 | P a g e

Determine and list the number of subsets that can be form from the set X, where X = {a, b, c}

Since X has 3 elements, the number of subsets that can be formed is 23 = 2 × 2 × 2 = 8 subsets

The subsets are:

No elements: {∅}

One element: {a}, {b}, {c}

Two elements: {a, b}, {a, c}, {b, c}

Three Elements: {a, b, c}

The subsets are: {{∅}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}

Points to note:

First use the formula to determine the number of subsets.

The empty set and the set itself are subsets.

These two sets will always be subsets of any given set.

Always form subsets in a systematic way starting with the set that has zero objects followed by

the set with one object followed by the set with two objects followed by the set with 3 objects

and so on.

Equal Sets

Two sets are said to be equal if and only if they contain the same objects. The order in which the

objects are placed does not matter.

Ex. A = {1, 2, 3, 4} B = {2, 1, 4, 3}

All the elements in A are in B and all the elements in B are also in A.

Since 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴, then A = B

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

7 | P a g e

It is observed by inspection that A is a subset of B and B is a subset of A.

Since both are subsets of each other, they are equal.

Therefore a necessary condition for two sets to be equal is they have to be subsets of each other.

Cardinality

The cardinality of a set represents the number of objects that can be found in a set by counting.

Let Z be the set of vowels.

Z = { a, e, i, o, u}

The cardinality of Z, written as n(Z) = 5.

Let T be the set of constants. Using set notation, state the set T and hence find the cardinality of

T.

T = { }

n(T) =

Equivalent Sets

Two sets are said to be equivalent if their cardinality are equal.

Ex. M = {1, 2, 3, 4} N = {a, b, c, d}

n(M)=4 n(N)=4

Since the cardinality of M and N are equal, M & N are equivalent sets.

n(Z) means the

number of

elements in Z

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

8 | P a g e

Compliment of a Set

Let A be a set. The compliment of the set A written as 𝐴’ represents all the objects that are not in

A.

Ex. Let U be the first 10 counting numbers.

U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

Let A be the first 5 even numbers.

A = {2, 4, 6, 8, 10}

Therefore A’ = {1, 3, 5, 7, 9}

Union

Consider the union of a bride and groom. The bride represents one set and the groom represents

another set. At the marriage ceremony, both the bride and groom come together with their

respective families.

In set theory, the union of two sets is just this.

If A and B are two sets, the union of A and B written as 𝑨 ∪ 𝑩 is the set that contains all the

elements found in A and B.

Let A = {1, 2, 3, 4} and let B = {a, b, c, d}

The union of A and B, 𝑨 ∪ 𝑩 = {𝟏 , 𝟐, 𝟑, 𝟒, 𝑎, 𝑏, 𝑐, 𝑑}

The objects 1, 2,

3, 4 are found in

the set A

The objects a, b,

c, d are found in

the set B

𝐴 ∪ 𝐵 represents the

combination of

elements belonging to

both A and B

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

9 | P a g e

1. Let X = {2, 4, 6, 8, 10} and Y = {1, 3, 5, 7, 9}. Using set notation, state:

a. The cardinality of X

b. The cardinality of Y

c. The union of X and Y

d. If X and Y are equivalent or equal

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

10 | P a g e

Intersection

Consider a boy and girl who wants to be with each other in a relationship. The boy will have his

likes and the girl will have her likes. The common things they both like together would represent

the intersection of their likes.

Therefore the intersection of two sets can be stated as:

If A and B are two sets, the intersection of A and B written as 𝐴 ∩ 𝐵, represents the objects that

are common to both A and B, ie, if it can be found in A, it can be found in B and if it can be

found in B, it can be found in A.

Let A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6}

The intersection of A and B, 𝐴 ∩ 𝐵 = { }

The objects

2, 4, 6 can be

found in the

set A

The objects 2, 4, 6 are

common to both A and B,

therefore they will be found

in the intersection

The objects 2, 4,

6 can be found in

the set B

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

11 | P a g e

1. Let X = { 5, 10, 15, 20} and Y = {10, 20, 30}. Using set notation, state:

a. The cardinality of X

b. The cardinality of Y

c. The union of X and Y

d. The intersection of X and Y

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

12 | P a g e

Venn Diagrams

A Venn diagram is a pictorial representation of sets. A rectangle is used to represent the

universe, U and circles are used to represents the sets themselves.

1. In the Venn diagram the shaded circle represents the objects that belong to A and the

unshaded region represents the objects that do not belong to A.

2. The shaded region represents the objects that belong to A and also belong to B, ie, A is

contained in B. A is a subset of B or 𝐴 ⊂ 𝐵 .

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

13 | P a g e

3. The shaded region represents the intersection or 𝐴 ∩ 𝐵.

4. The shaded region represents the union of A and B or 𝐴 ∪ 𝐵.

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

14 | P a g e

5. The shaded region represents the objects that belong to A only or 𝐴 ∩ 𝐵′

6. The shaded region represents the objects that belong to B only or 𝐴′ ∩ 𝐵

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

15 | P a g e

Consider two parties that have the following refreshments:

Party A = {Chips, Cookies, Cake, Soft Drink, Juice}

Party B = {Soft Drink, Cutters, Chips, Peanuts}

Both Party A & Party B have something in common;

A ∩ 𝐵 = {Chips, Soft Drink}

If we were to determine how was Party A unique, such that Party B does not have the objects in

Party A, this would just be the set: {Cookies, Cake, Juice}

It is important to note that these objects that are found in A AND not in B.

Using set notation, this translates to A ∩ 𝐵′.

Likewise, Party B is unique in such a way such that Party A does not have the objects in party B,

this is the set {Cutters, Peanuts}.

These are the objects that are found in B AND are not in A.

Using set notation, this translates to B ∩ 𝐴′.

A ∩ 𝐵′ = { }

𝐵 ∩ 𝐴′ = { }

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

16 | P a g e

Determine the cardinality of:

The set A n(A) =

The set B n(B) =

The set A ∩ 𝐵 n(A ∩ 𝐵) =

The set A ∩ 𝐵′ n(A ∩ 𝐵′) =

The set B ∩ 𝐴′ n(B ∩ 𝐴′) =

Next find 𝐴 ∪ 𝐵 = {Chips, Cookies, Soft Drink, Cutters, Peanuts, Juice, Cake}

n(𝐴 ∪ 𝐵) =

Note that n(A) + n(B) = 5 + 4 = 9

Therefore, n(A ∩ 𝐵) + n(A ∩ 𝐵′) + n(B ∩ 𝐴′) = 2 + 3 + 2 = 7

Points to note:

n(A ∩ 𝐵′) = n(A) - n(A ∩ 𝐵) = 5 – 2 = 3

n(B ∩ 𝐴′) = n(B) - n(A ∩ 𝐵) = 4 – 2 = 2

A general rule to follow is given as follows:

𝑛(𝐴 ∪ 𝐵) = 𝑛(𝐴) + 𝑛(𝐵) − 𝑛(𝐴 ∩ 𝐵)

Using the example before:

𝑛(𝐴 ∪ 𝐵) = 5 + 4 – 2 = 9 – 2 = 7

𝑛(𝐴 ∪ 𝐵) = 7

This is wrong since the total number of

objects in A and B is 7 (two objects are

repeated)

This represents the total number

of objects put together

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

17 | P a g e

May/June 2014

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

18 | P a g e

January 2014

WOODBROOK SECONDARY SCHOOL

MATHEMATICS

SET THEORY

FORM 4

19 | P a g e

May/June 2013