Mathematical content on Apronus.com is presented in Math ASCII Notation which can be properly displayed by all Web browsers because it uses only the basic set of characters found on all keyboards and in all fonts.

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A,B c X

1) A c X\B <=> ???

2) A c X\B <=> ???

1) A c X\B <=> ???

2) A c X\B <=> ???

1) A c X\B <=> B c X\A

2) A c X\B <=> B n A = O

2) A c X\B <=> B n A = O

A,B c X

1) A n (X\B) = O <=> ???

2) A n (X\B) = O <=> ???

1) A n (X\B) = O <=> ???

2) A n (X\B) = O <=> ???

1) A n (X\B) = O <=> A c B

2) A n (X\B) = O <=> X\B c X\A

2) A n (X\B) = O <=> X\B c X\A

A,B c X

1) A c B <=> ???

2) A c B <=> ???

3) A c B <=> ???

1) A c B <=> ???

2) A c B <=> ???

3) A c B <=> ???

1) A c B <=> X\B c X\A

2) A c B <=> A n (X\B) = O

3) A c B <=> A u B = B

2) A c B <=> A n (X\B) = O

3) A c B <=> A u B = B

A,B c X

A \ (X\B) = ???

A \ (X\B) = ???

A \ (X\B) = A n B = B n A = B \ (X\A)

Let X = {x:|N->|C : +(n=1 to oo) |x(n)| < oo}.

If x:-X, let g(x) = +(n=1 to oo) |x(n)|.

Is g a norm on X?

If x:-X, let g(x) = +(n=1 to oo) |x(n)|.

Is g a norm on X?

Yes.

Does this series converge?

1/n * (-1)^n * i^( n*(n+1) )

1/n * (-1)^n * i^( n*(n+1) )

YES.

a[n] = (-1)^n * i^( n*(n+1) )

a[1]=1 a[2]=-1 a[3]=-1 a[4]=1

a[n+4] = a[n]

Hence the series a[n] is bounded.

a[n] = (-1)^n * i^( n*(n+1) )

a[1]=1 a[2]=-1 a[3]=-1 a[4]=1

a[n+4] = a[n]

Hence the series a[n] is bounded.

Let D be an open subset of |R^2.

Let g : D -> |R. Suppose that the partial derivatives gx, gy both exist on D.

Let (a,b) be a point in D.

Suppose that /\(x,y):-D g(x,y) <= g(a,b).

What can we conclude?

Let g : D -> |R. Suppose that the partial derivatives gx, gy both exist on D.

Let (a,b) be a point in D.

Suppose that /\(x,y):-D g(x,y) <= g(a,b).

What can we conclude?

gx(a,b) = gy(a,b) = 0

Prove that

/\x:-|R \/k:-|Z [ -PI < x + k*2*PI <= PI ]

/\x:-|R \/k:-|Z [ -PI < x + k*2*PI <= PI ]

hint: k-1 < (x-PI)/(2*PI) <= k

page 62 in OLDTIMER

page 62 in OLDTIMER

-PI <= x,y <= PI

cos(x) = cos(y)

What can we conclude?

cos(x) = cos(y)

What can we conclude?

x = -y or x = y

Let x,y be real numbers such that cos(x)=cos(y).

What can we conclude?

What can we conclude?

\/k:-|Z x = y + k*2*PI or x = -y + k*2*PI

page 62 in OLDTIMER

page 62 in OLDTIMER