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.

The purpose of these pages is to demonstrate the power of the Math ASCII Notation. In principle, it can be used to write mathematical content of any complexity. In practice, its limits can be seen when trying to write complicated formulas (containing, for example, variables with many indexes or multiple integrals).

Despite its limitations the Math ASCII Notation has much expressive power, as can be seen from browsing through these pages.

Is it possible that z is not real and sin(z)=-1 ?

NO.

sin(z)=-1 ==> z is real

sin(z)=-1 ==> z is real

Let A[n] c X for all natural n.

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

Prove that for all x:-X

/\x:-X 1(lim_sup A[n])(x) = ???

If AcX, then let 1(A) denote the characteristic function of the set A defined on X.

Prove that for all x:-X

/\x:-X 1(lim_sup A[n])(x) = ???

/\x:-X 1(lim_sup A[n])(x) = lim_sup 1(A[n])(x)

The proof is very easy.

The proof is very easy.

Is it possible that z is not real and sin(z) is real?

YES.

cos(Re(z))=0 ==> sin(z) is real

sin(pi/2 + i) is real

cos(Re(z))=0 ==> sin(z) is real

sin(pi/2 + i) is real

Is it possible that z is not real and cos(z) is real?

YES.

cos(i) :- |R

sin(Re(z))=0 ==> cos(z) is real

cos(i) :- |R

sin(Re(z))=0 ==> cos(z) is real

Describe the set {z:-C : cos(z) is real}.

R u {z:-C : Re(z) = k*pi, where k is an integer}

cos(z) is real <=> [ z is real or sin(Re(z))=0 ]

page 27 in OLDTIMER

cos(z) is real <=> [ z is real or sin(Re(z))=0 ]

page 27 in OLDTIMER

Describe the set {z:-C : sin(z) is real}.

R u {z:-C : Re(z) = pi/2 * (2*k+1), where k is an integer}

sin(z) is real <=> [ z is real or cos(Re(z))=0 ]

page 27 in OLDTIMER

sin(z) is real <=> [ z is real or cos(Re(z))=0 ]

page 27 in OLDTIMER

Describe the set {z:-C : Re( cos(z) ) = 0}.

Re( cos(z) ) = 0 <=> cos( Re(z) ) = 0

page 27 in OLDTIMER

page 27 in OLDTIMER

Describe the set {z:-C : Re( sin(z) ) = 0}.

Re( sin(z) ) = 0 <=> sin( Re(z) ) = 0

When cos(z) is purely imaginary, what can we infer about sin(z)?

sin(z) is real

page 27 in OLDTIMER

page 27 in OLDTIMER

When sin(z) is purely imaginary, what can we infer about cos(z)?

cos(z) is real

Follow the downward chain of reasoning:

1) sin(z) is purely imaginary

2) Re(sin(z)) = 0

3) sin(Re(z)) = 0

4) cos(z) is real

Follow the downward chain of reasoning:

1) sin(z) is purely imaginary

2) Re(sin(z)) = 0

3) sin(Re(z)) = 0

4) cos(z) is real