Practice library
Problems
1Tomorrow's weather forecast consists of the following three claims: it will be cloudy, or it will snow, or the wind will blow; if it is cloudy and snowing, then the wind will blow; if the wind does not …Open2Let \(A\), \(B\) and \(C\) be finite sets whose sizes satisfy \[ |A \triangle C| + |B \triangle C| = |A \triangle B|. \] Prove that \(C\) is then trapped between the intersection and the union of the other …3Consider the set of five Serbian words (written in the Latin alphabet) \[ X = \{\ \text{aca},\ \text{konac},\ \text{lopte},\ \text{loto},\ \text{prst}\ \}, \] and define two relations on \(X\): for words …4Let \(S = \{s, i, c, g\}\). a) How many relations on \(S\) are not symmetric? b) How many antisymmetric relations are there on \(S\)?5Write each of the numbers \(1, 2, 3, \dots, 9\) into exactly one of the nine shapes in the figure - odd numbers into the triangles, even numbers into the squares - so that all \(12\) of the inequality …6At a round table sit \(2014\) people. Each of them either always tells the truth or always lies. Every single person at the table said the following sentence: "Apart from me and my two immediate neighbours, …7Let \(A\) and \(B\) be non-empty sets, neither of which is a subset of the other. For a natural number \(n\) consider the equality \[ \underbrace{A \setminus \bigl(B \setminus (A \setminus (B \setminus \cdots))\bigr)}_{n \text{ sets}} \;=\; \underbrace{A \mathbin{\triangle} \bigl(B \mathbin{\triangle} (A \mathbin{\triangle} (B \mathbin{\triangle} \cdots))\bigr)}_{n \text{ sets}} \] …8Two people are talking. Person \(A\) says: "If we win at football, we will also win at basketball." Person \(B\) says: "If we do not win at basketball, we will win at football." Person \(C\) remarks: "At …9Find all three-element sets \(A\) with the following two properties: (i) the set \(A\) has at least two elements in common with its power set \(\mathcal{P}(A)\); (ii) \(3 \in A\). (It is understood that …
Showing 9 of 651 - problem statements are free for everyone.