Notes on Set - Mathematical Structures | MAT 300, Study notes of Mathematics

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• Chapter

Definition 1Sets

A set is an unordered collection of objects.

Definition 2

Objects in a set are called elements or members

Definition 3of the set. A set is said to contain its elements.

Two sets are equal if they contain the same ele-

Definition 4ments.

The set

A

is a subset of

B

if and only if every

element of

A

is also an element of

B

We will denote this by

A

B

Russell’s Paradox

Let

A

X

X /

X

If

A

A

then by definition of

A

A /

A

If

A /

A

then by definition of

A , A ∈ A.

Operations on Sets

Union:

Let

A

B

be sets. Then

A

B = { x | x ∈ A

x

∈

B

} .

Intersection:

Let

A

B

be sets. Then

A ∩ B = { x | x ∈ A

x

∈

B

Difference:

Let

A

B

be sets. Then

A \ B = { x | x ∈ A ∧ x 6

B

Complement:

Let

U

be the universal set.

The complement

of the set

A , A ′ = U \ A.

Theorem 5Commutative laws

For any sets

A, B

A ∩ B = B ∩

A,

A ∪ B = B ∪

A.

Theorem 6Distributive laws

For any sets

A, B, C

A

B

C

A

B

( A ∪ C ) ,

A

B

C

A

B

( A ∩ C ).

Theorem 7De Morgan laws

For any subsets

A, B

of a universal set

U

( A ∩ B ) ′ = A ′ ∪ B ′

( A ∪ B ) ′ = B ′ ∩ A ′

Theorem 8The complementation and self-inverse laws

For any subset

A

of a universal set

U

A

A

′

( A ′ ) ′ =

A.

Theorem 9

For any subsets

A, B, C, D

A × ( B ∪ C

A

×

B

A

×

C

A × ( B ∩ C

A

×

B

A

×

C

A

×

B

C

×

D

A

C

×

B

D

( A × B ) ∪ ( C × D ) ⊆ ( A ∪

C

) × ( B ∪ D ).

A

B

C

D

If

F

is a family of sets then

⋃

F^

a |∃

F (^) ∈F

(^) a

∈

F (^) } ,

⋂

F^

a |∀

F (^) ∈F

(^) a

∈

F (^) } .

Theorem 10

Let

{ A i | i ∈ I }

be an indexed family of sets.

For

every

i 0 ∈ I , A i 0 ⊆

i∈ ⋃

I A

i , i∈ ⋂

I A i ⊆ A i 0.

Theorem 11

Let

A

i | i ∈

I }

be an indexed family of sets and let

B

be a set.

• B ∪ ⋃ i ∈ I A i = ⋃ i ∈ I

B ∪ A i ) ,

• B ∪ ⋂ i ∈ I A i = ⋂ i ∈ I

B ∪ A i ) ,

• B ∩ ⋃ i ∈ I A i = ⋃ i ∈ I

B ∩ A i ) ,

• B ∩ ⋂ i ∈ I A i = ⋂ i ∈ I

B ∩ A i ) ,

• ( ⋃ i ∈ I A i ) ′ = ⋂ i ∈ I A

i ′

• ( ⋂ i ∈ I A i ) ′ = ⋃ i ∈ I A

i ′

, a

] =

{ x ∈ R | x ≤ a }

a,

(^) ∞

x

∈

R

| a < x

[

a,

(^) ∞

{ x ∈ R | a ≤ x }

R

Absolute value

For any real number

a , | a | = √ a 2.

For any real numbers

a, b

a

· (^) b | =

a | · |

b | .

Triangle Inequality:

For any real numbers

a, b,

a

b | ≤ |

a | + | b |.

f (^) ( x ) | ≤

M

if and only if

M

f (^) ( x )

and

f (^) ( x ) ≤

M

f (^) (

x

) | ≥

M

if and only if

M

f (^) (

x

)

or

f (^) (

x

)

≤ −

M