Sets, Membership & Inclusion

(1)

Sets, Membership & Inclusion

European section, Season 01

(2)

Set

A set is a collection of distinct objects.

The elements or members of a set can be anything: numbers, people, letters of the alphabet, and so on.

Sets are conventionally denoted with capital letters.

European section, Season 01 Sets, Membership & Inclusion

(3)

Describing sets

There are two ways of describing a set.

(4)

Describing sets

One way is by definition, using a description :

A is the set whose elements are the first four positive integers.

B is the set of colors of the French flag.

(5)

Describing sets

One way is by definition, using a description :

A is the set whose elements are the first four positive integers.

B is the set of colors of the French flag.

The second way is by listing each member of the set. An

extensional definition is denoted by enclosing the list of members in brackets:

A = {4, 2, 1, 3}

B = {blue, white, red}

(6)

Membership

If a is a member of B, this is denoted a∈B, while if c is not a member of B then c ∈/ B.

(7)

Membership

For example, if A = {4, 2, 1, 3}, then 2∈A but 5∈/A.

(8)

Membership

For example, if A = {4, 2, 1, 3}, then 2∈A but 5∈/A.

×

a b

c

d e

f

g A

According to the Venn diagram, d ∈/ A whereas a∈A

(9)

Inclusion or Containment

If every member of set A is also a member of set B, then A is said to be a subset of B, written A⊂B (also pronounced A is contained in B).

(10)

Inclusion or Containment

B

A

(11)

Inclusion or Containment

B

A

A is a subset of B ; B is a superset of A.

A is contained in B ; B includes A or B contains A.

Sets A and B are equal if and only if they have precisely the same elements.

(12)

Special sets

(13)

Special sets

N,

(14)

Special sets

N, denoting the set of all natural numbers :

(15)

Special sets

N, denoting the set of all natural numbers : N={1,2,3, . . .}.

(16)

Special sets

N, denoting the set of all natural numbers : N={1,2,3, . . .}. Z,

(17)

Special sets

N, denoting the set of all natural numbers : N={1,2,3, . . .}. Z, denoting the set of all integers (whether positive, negative or zero):

(18)

Special sets

N, denoting the set of all natural numbers : N={1,2,3, . . .}. Z, denoting the set of all integers (whether positive, negative or zero): Z={. . . ,−2,−1,0,1,2, . . .}.

(19)

Special sets

Q,

(20)

Special sets

Q, denoting the set of all rational numbers (that is, the set of all fractions):

(21)

Special sets

Q, denoting the set of all rational numbers (that is, the set of all fractions): Q={^ab :a,b∈Z,b6=0}.

(22)

Special sets

Q, denoting the set of all rational numbers (that is, the set of all fractions): Q={^ab :a,b∈Z,b6=0}.

R, denoting the set of all real numbers. This set includes all rational numbers, together with all irrational numbers (that is, numbers which cannot be rewritten as fractions, such asπ,√

2, and so on).

(23)

Basic operations : Union

Two sets can be "added" together. The union of A and B, denoted by A∪B, is the set of all things which are members of either A or B.

Examples:

{1, 2}∪{red, white} =

(24)

Basic operations : Union

Examples:

{1, 2}∪{red, white} ={1, 2, red, white}.

(25)

Basic operations : Union

Examples:

{1, 2, green}∪{red, white, green} =

(26)

Basic operations : Union

Examples:

{1, 2, green}∪{red, white, green} ={1, 2, red, white, green}.

(27)

Basic operations : Union

Examples:

{1, 2}∪{1, 2} =

(28)

Basic operations : Union

Examples:

{1, 2}∪{1, 2} = {1, 2}.

(29)

Basic operations : Union

Examples:

{1, 2}∪{1, 2} = {1, 2}.

Venn’s representation :

(30)

Basic operations : Union

Examples:

{1, 2}∪{1, 2} = {1, 2}.

(31)

Basic operations : Intersection

The intersection of A and B, denoted by A∩B, is the set of all things which are members of both A and B.

(32)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =

(33)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

(34)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2, green}∩{red, white, green} =

(35)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2, green}∩{red, white, green} ={green}.

(36)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2}∩{1, 2} =

(37)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2}∩{1, 2} ={1, 2}.

(38)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2}∩{1, 2} ={1, 2}.

(39)

Basic operations : Intersection

Examples:

{1, 2}∩{red, white} =∅.

{1, 2}∩{1, 2} ={1, 2}.

(40)

Basic operations : complement

The complement of A is the set of all elements in the universe but not in A. We denote it A.

(41)

Basic operations : complement

Ω A

(42)

Basic operations : complement

Ω A

A