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Venn Diagrams A Venn diagram is a diagram that shows all possible logical relations between a finite collection of different sets. They can be used to illustrate simple set relationships in probability, logic, and statistics. Before we can effectively use a Venn diagram, we must first understand Sets. Think of a Set as a well-defined collection of objects or elements in which is it possible to determine if a given object is included in the collection. Sets are often labeled with a letter such as A = {1, 2, 3}. Example: U = {1,2,3,4,5,6,7,8,9, 10, 11}, A = {1,3,5,7} and B = {2,4,5,7,9,11} Find each set. (a) A ∩ B Solution: Set A ∩ B contains the elements that are found in both set A and B. Or in other words, their intersection. A ∩ B = {5,7} Now that we understand how a set works, let’s apply this knowledge to the use of a Venn diagram. Example: A researcher collecting data on 100 households finds that 76 have cats; 21 have dogs; and 12 have both dogs and cats. The researcher wants to answer the following questions. (a) How many do not have a cat? (b) How many have neither a cat nor a dog? (c) How many have a dog but not a cat? Solution: A Venn diagram will help us sort out this information. First, put the number 12 in the region common to both a cat and a dog, because 12 households have both. 1 U Cats 12 Dogs Of the 21 with a dog, 21 – 12 = 9 households with no cat, so we put 9 in the region for a dog but no cat. Similarly, 76 – 12 = 64 households with a cat but not a dog, so we put 64 in that region. Finally, the diagram shows that 100 – 64 – 12 – 9 = 15 households who have neither a dog nor a cat. Now we can answer the questions. (a) 15 + 9 = 24 do not have a cat. (b) 15 have neither. (c) 9 have a dog but not a cat. U = universal set (all households surveyed) Cats 64 12 Dogs 9 15 2