Pamela Leutwyler. definition a set is a collection of things – a set of dishes a set of clothing a set of chess pieces

Pamela Leutwyler

definition

a set is a collection of things –

a set of dishes

a set of clothing

a set of chess pieces

In mathematics, the word “set” refers to

“a well defined collection.”

The “collection of all short people” is not well defined (not a set)

The “collection of all people less than 4 feet tall” is well defined (a set)

notation

a b c d e

this collection of letters forms a set

a b c d e

separate the members of the set with commas

, , , ,

surround the members of the set with braces

{ }

a b c d e, , , ,{ }

It is conventional to use an upper case letter to name a set.

S =

a b c d e, , , ,{ }S =

this symbol means “ is a member of ”

a b c d e, , , ,{ }S =


“d is a member of the set S”

“d S”

is written

a b c d e, , , ,{ }S =

this symbol means “ is a member of ” not

a b c d e, , , ,{ }S =


“h is not a member of the set S”

“h S”

is written

not

a b c d e, , , ,{ }S =

b c e, ,{ }R =

a b c d e, , , ,{ }S =

b c e, ,{ }R =

the set R is related to the set S

a b c d e, , , ,{ }S =

b c e, ,{ }R =

the set R is related to the set S

every element of R is also an element of S

a b c d e, , , ,{ }S =

b c e, ,{ }R =

R is called a SUBSET of S

a b c d e, , , ,{ }S =

b c e, ,{ }R =

R is called a SUBSET of S

denoted

R S

the set of CANARIES is a subset of the set of BIRDS

the set of WOMEN is a subset of the set of HUMANS

the set of all COUNTING NUMBERS is called the set of NATURAL NUMBERS and is denoted “N”

N = { 1, 2, 3, 4, 5, 6, 7, 8, … }

the set of all COUNTING NUMBERS is called the set of NATURAL NUMBERS and is denoted “N”

N = { 1, 2, 3, 4, 5, 6, 7, 8, … }

the set of EVEN counting numbers is a subset of N

E N

two sets are said to be EQUAL if and only if theycontain the same members.

{1, 2, 3 } = { 2, 1, 3 }

in arithmetic we use the symbols: , , ,

greater than

greater than OR equal to

less than

less than OR equal to

If your age is less than 16, you cannot drive legally a 16

If your age is greater than or equal to 16, you can drive legally a 16

We use similar notation in set theory.

Suppose we know that every member of a set A is also a member of a set C.

We use similar notation in set theory.

Suppose we know that every member of a set A is also a member of a set C.

If it is possible that A is equal to C, we write

A C

example: A = the set of people enrolled in this class who will get an A in this class

C = the set of people enrolled in this class

A C B D

If you know that every member of Ais also a member of C then thissymbol is always correct

If you know that every member of Bis also a member of D and you also knowthat there is at least one member of D that is not in B, you can use this symbol.

In this case, B is called a PROPER SUBSET of D.

a set that has no members is called “EMPTY” andis represented with this symbol:

example: the set of all elephants who can sing opera in French =

the set that contains everything (or everything that you are talking about) is called the “UNIVERSE” andis represented by the symbol:

U

if you are solving algebraic equations then U = the set of real numbers

if you are a doctor studying the correlations of smoking with heart disease thenU = the set of people in your sample population

for any discussion of sets, the universe must be defined.

Sometimes it is difficult or impossibleto list all of the members of a set. We can use words to describe such sets:

the set of all people enrolled in this class

the set of all counting numbersthat are greater than 5

the set of all members of the US senate

“set builder notation” is the most efficient and accurate way to describethese sets. A variable symbol is usedto designate an unspecified member of the Universe:




= { x

“set builder notation” is the most efficient and accurate way to describethese sets. A variable symbol is usedto designate an unspecified member of the Universe:this slash is usually read “such that”




= { x /

“set builder notation” is the most efficient and accurate way to describethese sets. A variable symbol is usedto designate an unspecified member of the Universe:this slash is usually read “such that”the sentence gives the condition that defines membership.




= { x / x is a person enrolled in this class}

“set builder notation” is the most efficient and accurate way to describethese sets. A variable symbol is usedto designate an unspecified member of the Universe:this slash is usually read “such that”the sentence gives the condition that defines membership.




= { x / x is a person enrolled in this class}

= { x / x is a US senator }

= { x / x N and x 5 }

examples:

{ x / x N and 5 x 9 } = { 6, 7 ,8, 9 }

examples:

{ x / x N and 5 x 9 } = { 6, 7 ,8, 9 }

{ 2x / x N and 5 x 9 } = { 12, 14 ,16, 18 }

the number of members in a set Sis called the “cardinal number of S” denotes “n(S)”

A = { 1, 2, 3, 4 } n(A) = 4

V = { a, e, I, o, u } n(V) = 5

H = { p, q, r, s } n(H) = 4

the number of members in a set Sis called the “cardinal number of S” denotes “n(S)”

A = { 1, 2, 3, 4 } n(A) = 4

V = { a, e, I, o, u } n(V) = 5

H = { p, q, r, s } n(H) = 4

Two sets have the same cardinal number if and only if it is possible to establish a one to one correspondencebetween their members.

Two sets that have the same cardinal number are said to be EQUIVALENT.

E = the set of even counting numbers.

n( E ) = n( N )

E = { 2, 4, 6, 8,……………….. 2n,………}

N = { 1, 2, 3, 4, 5, 6, 7, 8, ……,n,……….}

operations

suppose:

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

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

suppose:

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

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

{ x / x A } =

suppose:

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

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

{ x / x A } = { 7, 8, 9, 10 }

this set is called the “complement” of A and is denoted “ A’ ”

A’ =

suppose:

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

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

{ x / x A AND x B } =

suppose:

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

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

{ x / x A AND x B } = { 2, 4, 6 }

this set is called the “intersection” of A with Band is denoted“A B”

A B =

suppose:

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

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

{ x / x A OR x B } =

suppose:

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

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

{ x / x A OR x B } =

{ }

1 2 83 4 5 6 2 4 6 10

suppose:

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

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

{ x / x A OR x B } =

{ 1, 2, 3, 4, 5, 6, 8, 10 }

this set is called the “union” of A with Band is denotedA B

A B =

suppose:

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

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

{ x / x A and x B } =

suppose:

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

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

{ x / x A and x B } = { 1, 3, 5 }

The members of B are removed from A.This is what remains.

suppose:

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

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

{ x / x A and x B } = { 1, 3, 5 }

this set is called “A minus B” and is denoted“A – B”

A - B =

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Pamela Leutwyler. definition a set is a collection of things – a set of dishes a set of clothing a set of chess pieces