Imagine you see a really huge party bus pulling out, an infinite bus with no seats. Instead everyone on board is identified by their unique name, which is an infinite sequence of \(0\)s and \(1\)s. The bus has every person named with every possible infinite sequence of \(0\)s and \(1\)s, someone is named \(00010000..00...\), someone else \(0101010101...\), and so on. Prove that this time you will not be able to accommodate all the new guests no matter how hard you try.
Prove the triangle inequality: in any triangle \(ABC\) the side \(AB < AC+ BC\).
In certain kingdom there are a lot of cities, it is known that all the distances between the cities are distinct. One morning one plane flew out of each city to the nearest city. Could it happen that in one city landed more than \(5\) planes?
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 \((A\cap B)\cup C\).
Let \(A,B\) and \(C\) be three sets. Prove that if we take an intersection \(A\cap B\) and intersect it with the set \(C\), we will get the same set as if we took an intersection of \(A\) with \(B\cap C\). Essentially, prove that it does not matter where to put the brackets in \((A\cap B)\cap C = A\cap (B\cap C)\). Draw a Venn diagram for the set \(A\cap B\cap C\).
Prove the same for the union \((A\cup B)\cup C = A\cup (B\cup C) = A\cup B\cup C\).
For three sets \(A,B,C\) prove that \(A - (B\cup C) = (A-B)\cap (A-C)\). Draw a Venn diagram for this set.