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Definitions:

Universal set :

I

Empty set:

∅

Union of sets

￼

A∪B={x:x∈Aorx∈B}

Intersection of sets

￼

A∩B={x:x∈Aandx∈B}

Complement

￼

A′={x∈I:x∉A}

Difference of sets

￼

A∖B={x:x∈Aandx∉B}

Cartesian product

￼

A×B={(x,y):x∈Aandy∈B}

Set identities involving union

Commutativity

￼

A∪B=B∪A

Associativity

￼

A∪(B∪C)=(A∪B)∪C

Idempotency

￼

A∪A=A

Set identities involving intersection

Commutativity

￼

A∩B=B∩A

Associativity

￼

A∩(B∩C)=(A∩B)∩C

Idempotency

￼

A∩A=A

Set identities involving union and intersection

Distributivity

￼

A∪(B∩C)=(A∪B)∩(A∪C)

￼

A∩(B∪C)=(A∩B)∪(A∩C)

Domination

￼

A∩∅=∅

￼

A∪I=I

Identity

￼

A∪∅=∅

￼

A∩I=A

Set identities involving union, intersection and complement

Complement of intersection and union

￼

A∪A′=I

￼

A∩A′=∅

De Morgan's laws

￼

(A∪B)′=A′∩B′

￼

(A∩B)′=A′∪B′

Set identities involving difference

￼

B∖A=B∖(A∪B)

￼

B∖A=B∩A′

￼

A∖A=∅

￼

(A∖B)∩C=(A∩C)∖(B∩C)

￼

A′=I∖A

Universal set :

I

Empty set:

∅

Union of sets

￼

A∪B={x:x∈Aorx∈B}

Intersection of sets

￼

A∩B={x:x∈Aandx∈B}

Complement

￼

A′={x∈I:x∉A}

Difference of sets

￼

A∖B={x:x∈Aandx∉B}

Cartesian product

￼

A×B={(x,y):x∈Aandy∈B}

Set identities involving union

Commutativity

￼

A∪B=B∪A

Associativity

￼

A∪(B∪C)=(A∪B)∪C

Idempotency

￼

A∪A=A

Set identities involving intersection

Commutativity

￼

A∩B=B∩A

Associativity

￼

A∩(B∩C)=(A∩B)∩C

Idempotency

￼

A∩A=A

Set identities involving union and intersection

Distributivity

￼

A∪(B∩C)=(A∪B)∩(A∪C)

￼

A∩(B∪C)=(A∩B)∪(A∩C)

Domination

￼

A∩∅=∅

￼

A∪I=I

Identity

￼

A∪∅=∅

￼

A∩I=A

Set identities involving union, intersection and complement

Complement of intersection and union

￼

A∪A′=I

￼

A∩A′=∅

De Morgan's laws

￼

(A∪B)′=A′∩B′

￼

(A∩B)′=A′∪B′

Set identities involving difference

￼

B∖A=B∖(A∪B)

￼

B∖A=B∩A′

￼

A∖A=∅

￼

(A∖B)∩C=(A∩C)∖(B∩C)

￼

A′=I∖A

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FORMULAES RELATED TO CUBES

after painting all sides of a

cube of N side ,if we cut the cube into some cubes if n side then

(1)-number of small cubes=[N÷n]^3

(2)-number if painted cubes in all sides,5 side,4side=0

(3)-number of cubes painted on 3 sides=8

(4)-number of cubes painted on 2 sides=[12(N-2n)]/n

(5)-number of cubes painted on 1 side=6[(N-2n)]^2/n

(6)- number of cubes not painted at any side=[(N-2n)/n]^3

after painting all sides of a

cube of N side ,if we cut the cube into some cubes if n side then

(1)-number of small cubes=[N÷n]^3

(2)-number if painted cubes in all sides,5 side,4side=0

(3)-number of cubes painted on 3 sides=8

(4)-number of cubes painted on 2 sides=[12(N-2n)]/n

(5)-number of cubes painted on 1 side=6[(N-2n)]^2/n

(6)- number of cubes not painted at any side=[(N-2n)/n]^3

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