For three sets \(A,B,C\) prove that \(A - (B\cup C) = (A-B)\cap (A-C)\). Draw a Venn diagram for this set.
For three sets \(A,B,C\) prove that \(A - (B\cap C) = (A-B)\cup (A-C)\). Draw a Venn diagram for this set.
How many subsets of \(\{1, 2, . . . , n\}\) are there of even size?
In how many ways can \(\{1, . . . , n\}\) be written as the union of two sets? Here, for example, \(\{1, 2, 3, 4\}\cup\{4, 5\}\) and \(\{4, 5\}\cup\{1, 2, 3, 4\}\) count as the same way of writing \(\{1, 2, 3, 4, 5\}\) as a union.
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).
Consider a set of natural numbers \(A\), consisting of all numbers divisible by \(6\), let \(B\) be the set of all natural numbers divisible by \(8\), and \(C\) be the set of all natural numbers divisible by \(12\). Describe the sets \(A\cup B\), \(A\cup B\cup C\), \(A\cap B\cap C\), \(A-(B\cap C)\).
Prove that the set of all finite subsets of natural numbers \(\mathbb{N}\) is countable. Then prove that the set of all subsets of natural numbers is not countable.
In a distant village, there are \(3\) houses and \(3\) wells. Inhabitants of each house want to have access to all \(3\) wells. Is it possible to build non-intersecting straight paths from each house to each well? All houses and well must be level (that is, none of them are higher up, like on a mountain, nor are any of them on lower ground, like in a valley).
Today you saw two infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. How do you accommodate everyone?
Now there are finitely many infinitely long buses with seats numbered as \(1,2,3,...\) carrying infinitely many guests each arriving at the full hotel. Now what do you do?