Weekly Challenge 12: Venn diagram fun

7053 /www/nrich/html/content/id/7053/ 1 2011-02-01T00:00:01 <mdoxml xmlns:ns0="http://nrich.maths.org/mdo" version="1.0"> <ul id="buttonBar"> <li> <a href="http://nrich.maths.org/675&part=">Warm-up problem</a> </li> <li> <a href="http://nrich.maths.org/7027&part=">Try this next</a> </li>  <li> <a href="http://en.wikipedia.org/wiki/Venn_diagram">Read all about it</a> </li> <li> <a href="http://nrich.maths.org/7051&part=solution">Last week's solution</a> </li> </ul>   <div>A Venn diagram is a way of representing all possible logical relationships between a collection of sets. The image shows a Venn diagram for three sets $A$, $B$ and $C$ <mdo:image align="right" alt="" height="159" src="Venn3.png" width="168" /> </div> How would you describe each of the seven regions in the diagram using unions $\cup$ and intersections $\cap$ of $A, B, C, A^c, B^c, C^c$ where the complements $A^c, B^c$ and $C^c$ of the sets $A, B$ and $C$ are the sets of elements not contained in $A, B$ and $C$ respectively relative to a universal set $A\cup B\cup C$ <div>Create a Venn diagram for 4 sets $A$, $B$, $C$ and $D$. Make sure that your diagram contains regions for all possible intersections and you might like to experiment to create a particularly pleasing diagram.</div> <div class="framework"> Did you know ... ? Venn diagrams are useful in many problems in logic, probability, computer science and set theory. As well as a useful tool, they are an area of study in themselves and much research has gone into the creation of particularly beautiful or symmetric Venn diagrams for larger numbers of sets.</div>   </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Consider the Venn diagram for 3 sets $A$, $B$ and $C$ with each of the $7$ regions labelled as follows <mdo:image height="159" width="168" alt="" src="Venn3Soln.png"></mdo:image> The set-theoretic representation of these regions is 1. $C^c\cap B^c$ 2. $A\cap B \cap C^c$ 3. $A^c \cap C^c$ 4. $A\cap C \cap B^c$ 5. $A\cap B\cap C$ 6. $B\cap C \cap A^c$ 7. $A^c \cap B^c$ A four-region diagram can be constructed by overlapping four rectangles (blue, pink, yellow, red) as follows: <mdo:image height="300" width="300" alt="" src="Venn4.png"></mdo:image> Note that there are $15$ distinct regions of the diagram, corresponding to $2^4-1$. If your diagram does not have exactly $15$ distinct regions then it is incorrect! </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Consider the Venn diagram for 3 sets $A$, $B$ and $C$ with each of the $7$ regions labelled as follows <mdo:image height="159" width="168" alt="" src="Venn3Soln.png"></mdo:image> The set-theoretic representation of these regions is 1. $C^c\cap B^c$ 2. $A\cap B \cap C^c$ 3. $A^c \cap C^c$ 4. $A\cap C \cap B^c$ 5. $A\cap B\cap C$ 6. $B\cap C \cap A^c$ 7. $A^c \cap B^c$ A four-region diagram can be constructed by overlapping four rectangles (blue, pink, yellow, red) as follows: <mdo:image height="300" width="300" alt="" src="Venn4.png"></mdo:image> Note that there are $15$ distinct regions of the diagram, corresponding to $2^4-1$. If your diagram does not have exactly $15$ distinct regions then it is incorrect! </mdoxml> 2 3 0 0 0 0 1 Weekly Challenge 12: Venn diagram fun A weekly challenge: these are shorter problems aimed at Post-16 students or enthusiastic younger students. Collections Weekly Challenge