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 true that

(A u B) + (C u D) c (A + C) u (B + D)

?

(A u B) + (C u D) c (A + C) u (B + D)

?

Yes.

Is it true that

(A n B) + (C n D) c (A + C) u (B + D)

?

(A n B) + (C n D) c (A + C) u (B + D)

?

Yes.

Is it true that

(A\B) + (C\D) c (A + C) u (B + D)

?

(A\B) + (C\D) c (A + C) u (B + D)

?

Yes.

Is it true that

(A n B) u (C n D) c (A u C) n (B u D)

?

(A n B) u (C n D) c (A u C) n (B u D)

?

Yes.

Is it true that

(A u C) n (B u D) c (A n B) u (C n D)

?

(A u C) n (B u D) c (A n B) u (C n D)

?

No.

Let A and D be empty sets.

Let B = C = {1}.

Let A and D be empty sets.

Let B = C = {1}.

Is it true that

(A + C) u (C + B) c (A + B)

?

(A + C) u (C + B) c (A + B)

?

No.

Let A and B be empty sets.

Let C be non-empty.

Let A and B be empty sets.

Let C be non-empty.

Is it true that

(A + C) u (B + D) c (A u B) + (C u D)

?

(A + C) u (B + D) c (A u B) + (C u D)

?

No.

1) Let A = D = {1}. Let B = C = {2}.

2) Let A = B = C = {1}. Let D = O.

3) A=D={1}, C=O

1) Let A = D = {1}. Let B = C = {2}.

2) Let A = B = C = {1}. Let D = O.

3) A=D={1}, C=O

Is it true that

(A + C) u (B + D) c (A n B) + (C n D)

?

(A + C) u (B + D) c (A n B) + (C n D)

?

No.

Let A = {1}, B = C = D = 0.

The right side is empty.

1 belongs to the left side.

Let A = {1}, B = C = D = 0.

The right side is empty.

1 belongs to the left side.

Is it true that

(A + C) u (B + D) c (A\B) + (C\D)

?

(A + C) u (B + D) c (A\B) + (C\D)

?

No.

Let C=D=0, A=B={1}.

Let C=D=0, A=B={1}.

A[n], B[n] are sequences of subsets of X.

lim_inf ( A[n] n B[n] ) c ( lim_inf A[n] ) n ( lim_inf B[n] )

Is this true?

lim_inf ( A[n] n B[n] ) c ( lim_inf A[n] ) n ( lim_inf B[n] )

Is this true?

Yes.

Even equality holds.

lim_inf ( A[n] n B[n] ) = ( lim_inf A[n] ) n ( lim_inf B[n] )

Even equality holds.

lim_inf ( A[n] n B[n] ) = ( lim_inf A[n] ) n ( lim_inf B[n] )