What is the difference between a set and a region in math

At this party, two sets are being combined, though it might turn out that there are some friends that were in both sets. The union of two sets contains all the elements contained in either set or both sets. The intersection of two sets contains only the elements that are in both sets. The complement of a set A contains everything that is not in the set A. Notice that in the example above, it would be hard to just ask for A c , since everything from the color fuchsia to puppies and peanut butter are included in the complement of the set.

For this reason, complements are usually only used with intersections, or when we have a universal set in place.

A universal set is a set that contains all the elements we are interested in. This would have to be defined by the context. A complement is relative to the universal set, so A c contains all the elements in the universal set that are not in A.

Grouping symbols can be used like they are with arithmetic — to force an order of operations. To visualize the interaction of sets, John Venn in thought to use overlapping circles, building on a similar idea used by Leonhard Euler in the eighteenth century.

These illustrations now called Venn Diagrams. A Venn diagram represents each set by a circle, usually drawn inside of a containing box representing the universal set. Overlapping areas indicate elements common to both sets. A c will contain all elements not in the set A. The elements in the outlined set are in sets H and F , but are not in set W. Often times we are interested in the number of items in a set or subset.

This is called the cardinality of the set. Sometimes we may be interested in the cardinality of the union or intersection of sets, but not know the actual elements of each set.

This is common in surveying. Suppose 20 report tea only, 80 report coffee only, 40 report both. How many people drink tea in the morning? How many people drink neither tea or coffee?

This question can most easily be answered by creating a Venn diagram. We can see that we can find the people who drink tea by adding those who drink only tea to those who drink both: 60 people. We can also see that those who drink neither are those not contained in the any of the three other groupings, so we can count those by subtracting from the cardinality of the universal set, How many people have used neither Twitter or Facebook? Am I correct that I understood the definition of domain to be an open, connected set?

Does every region have to be a domain? Why would this be a region? A region is a set whose interior is a domain and which is contained in the closure of its interior. Sign up to join this community. The best answers are voted up and rise to the top. Region and domains? Asked 11 years, 10 months ago. Active 7 years ago.

Viewed 8k times. Improve this question. Ben McKay Anonymous Anonymous 55 1 1 gold badge 4 4 silver badges 5 5 bronze badges. In a similar manner, there are several ways to create new sets from sets that have already been defined. In fact, we will form these new sets using the logical operators of conjunction and , disjunction or , and negation not.

These sets are examples of some of the most common set operations, which are given in the following definitions. That is,. However, it is also helpful to have a visual representation of sets. Venn diagrams are used to represent sets by circles or some other closed geometric shape drawn inside a rectangle. The four distinct regions in the diagram are numbered for reference purposes only. The numbers do not represent elements in a set. The following table describes the four regions in the diagram.

We can use these regions to represent other sets. For each of the following, draw a Venn diagram for two sets and shade the region that represent the specified set. In addition, describe the set using set builder notation. We need one more definition. One reason for the definition of proper subset is that each set is a subset of itself. For example, if. It is often very important to be able to describe precisely what it means to say that one set is not a subset of the other.

So when we negate this, we use an existential quantifier as follows:. Another way to look at this is to consider the following statement:.

For example,. This gives us the following test for set equality:. For each blank, include all symbols that result in a true statement. If none of these symbols makes a true statement, write nothing in the blank. In that preview activity, we restricted ourselves to using two sets. We can, of course, include more than two sets in a Venn diagram. In this diagram, there are eight distinct regions, and each region has a unique reference number.