Venn Diagrams
A Venn diagram is a type of diagram used in mathematics, logic, linguistics and many other fields. It is named after John Venn, who pioneered the use of such diagrams for logic in his 1881 book Symbolic Logic. The Venn diagram consists of two or more overlapping circles (or ellipses). The set of elements common to each circle is shown as an area common to both circles. Where the circles overlap, only one element can be present, so this area represents the union of the sets represented by the distinct circles. Let’s first write the set A in our old notation. We have curly braces 1, 5, 10, 2. So we know that A is equal to what? It’s equal to four. Okay. Here we’ve written the set by listing out its elements explicitly. Another way to write it is to write a big circle and put the elements inside floating around– 1, 5, 2 – these are just some elements of A. We just think of A, a bag with some things in it; there they are. Okay? And by the way, you can write this any way you want; so maybe this is the same as 1, 2, 10, 5; it doesn’t matter how you write it as long as everybody knows what it means. This technique for visualizing intersections can be used to illustrate things like intersections. Let’s say we have two sets with respective memberships A = 1, 2, 10 and 5, and B = 5, -7, 10 and 3. Both A and B intersect at 10 and 5 because they share them in common. We can demonstrate that by overlapping the two sets like this. Notice that the elements of B not found in A are -7 and 3. Similarly, we would be able to tell that C is disjoint from A because it has no elements in common with any other set. A intersect C is the empty set, B intersect C is the empty set. Okay. So that’s kind of neat. One formula that is always true involves the union of two sets. This formula is concerned with the count of elements in a set when there are other sets involved. When we consider the elements of the union of two sets, we must include those that are in either or both of the sets. The inclusion-exclusion formula states that the cardinality (size) of this set is equal to that of set A plus that of set B minus the number of elements in common between A and B. Let us first check to see if this is true. Working over here, we know the cardinality of A union B is equal to the sum of their individual cardinals. We can count this easily by totaling each set: A consists of 1, 2, 10, 5 and -7; B consists of 10, 5 and -7; A intersect B consists of only 10 and 5. We are essentially asking if 6 = 4+4-2; in fact, we find that this statement holds true. Let us erase this question mark and put in a checkmark instead indicating that our statement is correct. If we take the cardinality of set A, we count its elements-there are 10 elements. If we then take the cardinality of set B and count its elements, there are five elements. What’s wrong with saying that the cardinality of A union B equals the cardinality of A plus B’s cardinality? Well, we’ve double-counted; we’ve given ourselves too much credit because we counted 10 and 5 twice. So to account for that, we subtract one copy of 10 and one copy of 5 from our answer. Therefore, cardinality of A union B equals A’s cardinality plus B’s cardinality minus the number of elements that are common to both sets (in this case, two). Now that we have the Venn diagram, let us revisit our medical testing example. So let us remember that X was equal to all the people who took some exam; these are all the people, then, who did not have I believe we called it VBS for very bad syndrome-X intersect S: the people who had it. And so we draw a partition here: a line separates the healthy from the sick. Notice that H intersect S is empty: no one has both diagnoses. And so somehow we use that line to divide the two groups. [Graph][Graph][Graph] We can make another partition of X. where X is the set of people who tested positive for VBS. Some of these people will be relieved to find that they do not have VBS, while others will be distressed by their diagnosis. So on this side might be the people who tested negative, and on this side might be the people who tested positive. And notice the way I’ve drawn it: Venn diagrams are never a way to compute something, they’re a way to encode visual assumptions. Notice that H and N are not the same but share a lot of area, and P and S are not the same but share a lot of area. This figure illustrates an important concept in diagnostic testing, where the S and N sets are those whose test results indicate that they have a disease and those whose test results indicate that they do not have the disease. For example, the people in S but also in N are false negatives – they have the disease but their test tells them they do not. On the other hand, H intersecting P is a false positive: people who do not have the disease but whose tests indicate that they do. Ideally, a perfect diagnostic test would result in no intersection between S and N or H and P: this would ensure that everyone who does have a disease gets treatment for it and that those who do not have it avoid unnecessary anxiety associated with being falsely identified as having it.
