SET IDENTITIES

Sets; A,B,C

Universal set; I

Compliment;  A’

Proper set; A⊂B

Empty set; Φ

Union of set; A∪B

Intersection of sets; A∩B

Difference of sets; A/B

 

  • A⊂I
  • A⊂A
  • A=A if A⊂B and B⊂A
  • Empty set Φ⊂A
  • Union of sets; C=A∪B={x|x ∈ A or x ∈ B}sets1
  • Commutativity; A∪B=B∪A
  • Associativity; A∪(B∪C) = (A∪B)∪C
  • Intersection of sets; C = A∪B={x|x ∈ A and x∈B}set2
  • Commutativity ; A∩B =B∩A
  • Associativity ; A∩(B∩C)=(A∩B)∩C
  • Distributivity; A∪(B∩C)=(A∪B)∩(A∪C);   A∩(B∪C)=(A∩B)∪(A∩C)
  • Idempotency; A∪A=A;   A∩A=A
  • Domination ; A∩Φ=Φ ;   A∪I=I
  • Identity; A∪Φ=A;   A∩I=I
  • Complement; A’={x∈I|x∉I}
  • Complement of intersection and union; A∪A’=I;  A∩A’=Φ
  • De Morgan’s Laws; (A∪B)’=A’∩B’;   (A∩B)’=A’∪B’
  • Difference of sets; C=B/A-{x|x∈B and x∉A}

 

set3

  • B/A=B/(A∩B)
  • B/A=B∩A’
  • A/A=Φ
  • A/B=A if A∩B=Φ

set4

  • (A/B)∩C= (A∩C)/(B∩C)
  • A’=I/A
  • Cartesian Product C=A x B ={(x,y)|x∈A and Y∈B}

 

SETS OF NUMBERS

Natural numbers; N

Whole numbers;  Nο

Integers; Z

Positive integers; Z+

Negative integers; Z-

Rational numbers; Q

Real numbers; R

Complex numbers; C

 

  • Natural numbers,N = {1,2,3,4,……∞} (counting numbers)
  • Whole numbers, Nο= {0,1,2,3,4,……∞} (counting numbers and zero)
  • Integers; (whole numbers and their opposites and zero)
    • Z+= N= {1,2,3,….}
    • Z-= {…..,-3,-2,-1}
    • Z = Z- ∪{0}∪Z+ = {…..,-3,-2,-1,0,1,2,3,4,….}
  • Rational numbers; Q={x|x=a/b and a∈z and b∈z and b is not equal to zero} (repeating or terminating decimals)
  • Irrational numbers; Non repeating and non terminating decimals
  • Real numbers; union of rational and irrational numbers: R
  • Complex numbers; {x+iy|x∈R and y∈R} where i is the imaginary unit
  • N⊂Z⊂Q⊂R⊂C

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