SET IDENTITIES
Sets: P,Q,R
Universal Set I
Compliment:P^{'}
Proper subset: P\subset Q
Empty Set \phi
Union of sets: P\cup Q
Intersection of sets: P\cap Q
Difference of sets: P\setminus Q
1. P\subset I
2. P\subset P
3. P=Q if P\subset Q and Q\subset P
4. Empty Set
\phi \subset P
5. Union of Sets
R=P\cup Q=\{x|x\in P\:or\: x\in Q \}
6. Commutativity
P\cup Q=Q\cup P
7. Associative
P\cup (Q\cup R)=(P\cup Q)\cup R
8. Intersection of Sets
C=P\cap Q=\{x|x\in P\:and\: x\in Q \}
9. Commutativity
P\cap Q=Q\cup P
10. Associative
P\cap (Q\cap R)=(P\cap Q)\cap R
11. Distributivity
P\cup (Q\cap R)=(P\cup Q)\cap (P\cup R)
P\cap (Q\cup R)=(P\cap Q)\cup (P\cap R)
12. Idempotency
P\cap P=P
P\cup P=P
13. Domination
P\cap \phi=\phi
P\cup I=I
14. Identity
P\cup \phi=P
P\cap I=P
15. Compliment
P^{'}=\{x\in I|x\not\in P\}
16. Compliment of Intersection and Unions
P\cup P^{'}=I
P\cap P^{'}=\phi
17. De Morgan's Laws
(P\cup Q)^{'}=P^{'}\cap Q^{'}
(P\cap Q)^{'}=P^{'}\cup Q^{'}
18. Difference of Sets
R=Q\setminus P=\{x|x\in Q\: and \: x\not\in P\}
19. Q\setminus P=Q\setminus (P\cap Q)
20. Q\setminus P=Q\cap P^{'}
21. P\setminus P=\phi
22. P\setminus Q=P\:if\:P\cap Q=\phi
23. (P\setminus Q)\cap R=(P\cap R)\setminus (Q\cap R)
24. P^{'}=I\setminus P
25. Cartesian Product
R=P\times Q=\{(x,y)|x\in P\: and\: y\in Q\}
SETS OF NUMBERS
Natural Number:N
Whole Number:N_{0}
Integer:Z
Positive Integer:Z^{+}
Negative Integer:Z^{-}
Rational Integer:Q
Real Integer:R
Complex Numbers:C
26. Natural Numbers
Counting Numbers:N=\{1,2,3,...\}
27. Whole Numbers
Counting Numbers and zero:N_{0}=\{0,1,2,3,...\}
28. Integers
Whole numbers and their opposites and zero:
Z^{+}=N=\{1,2,3,...\},
Z^{-}=N=\{...,-3,-2,-1\},
Z=Z^{-}\cup \{0\}\cup Z^{+}=\{...,-3,-2,-1,0,1,2,3,...\}
29. Rational Numbers
Repeating or terminating decimals:
Q=\{x|x=\frac{a}{b}\: and\: a\in Z\: and\: b\in Z\:and\: \neq 0\}
30. Irrational Numbers
Nonrepeating and nonterminating decimals:
Q^{'}=\{x|x\neq\frac{a}{b}\: and\: a\in Z\: and\: b\in Z\:and\: \neq 0\}
31. Real Numbers
Union of rational and irrational numbers:
R=Q\cup Q^{'}
32. Complex Numbers
C=\{x+iy|x\in R\: and\: y\in R\},
where i is the imaginary unit.
33. N\subset Z\subset Q\subset R\subset C
BASIC IDENTITIES
Real Numbers:a,b,c
34. Additive Identity
a+0=a
35. Additive Inverse
a+(-a)=0
36. Commutative Of Addition
a+b=b+a
27. Associative Of Addition
(a+b)+c=a+(b+c)
38. Definition Of Subtraction
a-b=a+(-b)
39. Multiplicative Identity
a.1=a
40. Multiplicative Inverse
a.\frac{1}{a}=1,a\neq 0
41. Multiplication Times 0
a.0=0
42. Commutative Of Multiplication
a.b=b.a
43. Associative Of Multiplication
(a.b).c=a.(b.c)
44. Distributive Law
a(b+c)=ab+ac
45. Definition of Division
\frac{a}{b}=a.\frac{1}{b}
COMPLEX NUMBERS
Natural number:n
Imaginary Number:i
Complex Number:z
Real part:a,c
Imaginary part:bi,di
Modulus of a complex number:r,r_{1},r_{2}
Argument of a complex number:\theta,\theta_{1},\theta_{2}
46.
i^{1}=i\quad\: i^{5}=i \quad\quad i^{4n+1}=i\\ i^{2}=-1\quad i^{6}=-1 \quad i^{4n+2}=-1\\ i^{3}=-i\quad i^{7}=-i \quad i^{4n+3}=-i\\ i^{4}=1\quad\: i^{8}=1 \quad\quad i^{4n}=1
47. z=a+bi
48. Complex Plane
49. (a+bi)+(c+di)=(a+c)+(b+d)i
50. (a+bi)-(c+di)=(a-c)+(b-d)i
51. (a+bi)(c+di)=(ac-bd)+(ad+bc)i
52. \frac{a+bi}{c+di}=\frac{ac+bd}{c^{2}+d^{2}}+\frac{bc-ad}{c^{2}+d^{2}}i
53. Conjugate Complex Numbers
\overline{a+bi}=a-bi
54. a=rcos\theta ,b=rsin\theta
55. Polar Presentation of Complex Numbers
a+bi=r(cos\theta+isin\theta
56. Modulus and Argument of a complex Number
If a+bi is a complex number, then
r=\sqrt{a^{2}+b^{2}}(modulus),
\theta=arctan\frac{b}{a}(argument).
57. Product in Polar Representation
z_{1}z_{2}=r_{1}(cos\theta_{1}+isin\theta_{1}).r_{2}(cos\theta_{2}+isin\theta_{2})
=r_{1}r_{2}[cos(\theta_{1}+\theta_{2})+isin(\theta_{1}+\theta_{2})]
58. Conjugate Numbers in Polar Representation
\overline{r(cos\theta+isin\theta)}=r[cos(-\theta)+isin(-\theta)]
59. Inverse of a Complex Number in Polar Representation
\frac{1}{r(cos\theta+isin\theta)}=\frac{1}{r}[cos(-\theta)+isin(-\theta)]
60. Quotient In Polar Representation
\frac{z_{1}}{z_{2}}=\frac{r_{1}(cos\theta_{1}+isin\theta_{1})}{r_{2}(cos\theta_{2}+isin\theta_{2})}=\frac{r_{1}}{r_{2}}[cos(\theta_{1}-\theta_{2})+isin(\theta_{1}-\theta_{2})]
61. Power of a Complex Number
Z^{n}=[r(cos\theta+isin\theta)]^{n}=r^{n}[cos(n\theta)+isin(n\theta)]
62. Formula "DE Moivre"
(cos\theta+isin\theta)^{n}=cos(n\theta+isin(n\theta)
63. Nth Root of a Complex Number
\sqrt[n]{z}=\sqrt[n]{r(cos\theta+isin\theta)}=\sqrt[n]{r}(cos\frac{\theta+2\pi k}{n}+isin\frac{\theta+2\pi k}{n})
where k=0,1,2,...,n-1
64. Euler's Formula
e^{ix}=cosx+isinx