Set Theory

Definition of a set

Definition of a set

A collection of objects that share one or more well-clarified properties is called a set

What are the number sets?

N, Z, Q, R, C

Definition of the empty set (∅)

A set with no elements is called the empty set

Definition of a subset

B is called a subset of A if all elements of B are in A

Definition of a proper subset

B is called a proper subset of A if all elements of B are in A but not all elements of A are in B

Can be venn-diagrams be used as proof?

No

Definition of an union

AuB= {x|xeA or xeB}

What is the union of A={1,2} and B={4,5,6}

AuB= {1,2,4,5,6}

What are the properties of the union?

• Closure: yes

• Associativity: yes

• Neutral element: empty set

• Commutative: yes

Definition of the intersection

AnB= {y|yeA and yeB}

Properties of the intersection

• Closure: yes

• Associativity: yes

• Neutral element: none

• Commutative: yes

Definition of the difference (\)

A\B= {x|xeA and xe/B}

Properties of the difference

• Closure: yes

• Associativity: no

• Neutral element: empty set

• Commutative: no

Definition of the complement of a set (ε)

ε is given A’=ε\A is called the complement of A in ε

Properties of a complement of a set (ε)

• (A’)’= A

• A’nA= empty set

• A’uA= ε

Draw the proof for Laws of de Morgen (if correct type correct)

Correct

Definition of symmetric difference (Δ)

AΔB= (AuB)\(AnB)

Properties of symmetric difference

• Closure: yes

• Associativity: yes

• Neutral element: empty set

• Commutative: yes

Proof of A\B=AnB’ (if correct type correct)

correct

Proof for the distributivity laws (if correct type correct)

correct

Definition of the power set

A is a set P(A)={B|B is a subset of A} is called the power set of A

Definition of cardinality of a set

The cardinality of a set denotes the number of elements of the set

What are the infinite sets that are countable?

N, Z, Q

What are the infinite sets that are uncountable?

R, C