Problems

How would you describe the surface obtained by glueing the sides of the octagon as on the picture? Sides of the same colour are glued together in the same direction as shown.

Describe the surface which we can get if we start with a rectangular sheet of paper, make a cylinder by glueing the opposite sides in the same direction and then glue the other opposite sides of the paper band in the opposite direction as on the picture.

Describe the surface which we can get if we start with a rectangular sheet of paper, make a Moebius band by glueing the opposite sides in the opposite directions and then glue the other opposite sides of the paper band in the opposite direction as on the picture.

Scrooge McDuck has \(100\) golden coins on his office table. He wants to distribute them into \(10\) piles so that no two piles contain the same amount of coins. Moreover, no matter how you divide any of the piles into two smaller piles, among the resulting \(11\) piles there will be two with the same amount of coins. Find an example of how he could do that.

We can define the absolute value \(|x|\) of any real number \(x\) as follows. \(|x|=x\) if \(x\ge0\) and \(|x|=-x\) if \(x<0\). What are \(|3|\), \(|-4.3|\) and \(|0|\)?

Prove that \(|x|\ge0\).

Prove that \(|x|\ge x\). It may be helpful to compare each of \(|3|\), \(|-4.3|\) and \(|0|\) with \(3\), \(-4.3\) and \(0\) respectively.

Let \(A=\{1,2,3\}\) and \(B=\{2,4\}\) be two sets containing natural numbers. Find the sets: \(A\cup B\), \(A\cap B\), \(A-B\), \(B-A\).

Let \(A=\{1,2,3,4,5\}\) and \(B=\{2,4,5,7\}\) be two sets containing natural numbers. Find the sets: \(A\cup B\), \(A\cap B\), \(A-B\), \(B-A\).

Given three sets \(A,B,C\). Prove that if we take a union \(A\cup B\) and intersect it with the set \(C\), we will get the same set as if we took a union of \(A\cap C\) and \(B\cap C\). Essentially, prove that \((A\cup B)\cap C = (A\cap C)\cup (B\cap C)\).

\(A,B\) and \(C\) are three sets. Prove that if we take an intersection \(A\cap B\) and unite it with the set \(C\), we will get the same set as if we took an intersection of two unions \(A\cup C\) and \(B\cup C\). Essentially, prove that \((A\cap B)\cup C = (A\cup C)\cap (B\cup C)\). Draw a Venn diagram for the set \((A\cap B)\cup C\).