Set Operations Objectives At the end of this lesson you should be able to 1 2 3 4 Describe the different set operations Make and use a Venn diagram Understand basic facts about sets Understand deMorgan s Properties Background In this lesson you will begin to see how we can combine sets to create new sets This will prove very useful as we try to count the number of elements in a set If you are very familiar with sets you can probably skim through this lesson Howeve keep it in mind for reference as you work with sets We will use a convenient picture of a set called a Venn Diagram These diagrams are very easy to develop and will allow us to see many of the definitions to follow Venn Diagrams A Venn Diagram of a single set is simply a circle Suppose we want to discusses subsets of objects in this universal set U 9 We could use the roster method It is simple and requires no particular artistic skill Let S represent the set of objects in U that have a square corner Then the subset of U is exactly the three elements below S 9 A Venn Diagram The Venn Diagram is a very simple picture to draw 1 2 3 4 5 Draw a rectangle and label it as U for the Universal Set Draw a circle somewhere inside the rectangle and label it as S Draw the objects in the list for S inside the circle Everything else in U is outside the circle We sometimes add shading to make it easier yet to see what set we are focused on Thanks to the Venn Diagram we can now introduce a new term ComplEment of a Set written symbolically as depending on the text book you read In this course the usual symbol will be The goo spElling is to emphasize that this is not a COMPLiMENT such as My goodness you are good in math In the diagram above S is shaded while the white inside the rectangle is the complement of S For a set A the complement of A is all of the elements in the universal set U not in A Fact The complement of the complement of a set is the original set Arizona State University Department of Mathematics and Statistics 1 of 3 Set Operations Operations on Sets Set operations are critical to working with sets They can be used to symbolically describe a set when language begins to create confusion The set operations we use show us how to combine or unite sets union or how to identify commonality among sets intersection The result of both of these operations are sets themselves Union We can list all of the elements of two sets to create another set In doing so any duplications are ignored so that each element is mentioned exactly one time We need to eliminate duplications because the next step is to count the distinct uniquely listed elements The symbol for union is a u shape c sometimes called the cup Example Let s use the universal set of the numeric digits U 1 2 3 4 5 6 7 8 9 0 Notice that zero is listed last here As a set these numbers have no particular order Let A 1 2 4 8 and B 3 5 4 8 The union of the set A with B is symbolized by A c B 1 2 3 4 5 8 We kept only one of each symbol So while 4 0 A and 4 0 B we list 4 only once The same applies to 8 Example For same U let A 1 2 4 8 We can see that 3 5 6 7 9 0 The union of the set A with the set is A c It is U itself This is always true In other words the union of a set and its complement is always the universal set Fact A c U always and don t you forget it Let A 1 2 4 8 and C 1 3 5 6 7 9 0 The union of the sets A and C is again U itself However notice that C Fact For A f U and B f U A c B f U always That is if A is a subset of the universal set and B is a subset of the universal set then the union of A and B is also a subset of the universal set Fact If A c B i then both A i and B i That is if the union of A and B is empty then both A and B are empty Intersection We can find the common elements within two sets called the intersection Another way to think about it is that the operation of intersection is one where we state the overlap of two sets by listing its elements as a set Example Let A 1 2 4 8 and B 3 5 4 8 The intersection of the sets A and B is symbolized by A 1 B 4 8 Notice 4 0 A and 4 0 B but we list 4 only once Example Let A 1 2 4 8 and C 3 5 6 7 9 The intersection of the sets A and C is empty Hence A 1 C i 2 of 3 Arizona State University Department of Mathematics and Statistics Set Operations Definitions When any two sets have no elements in their intersection they are called disjoint When a group of sets each has nothing in common with any of the others in question they are pairwise disjoint Let A 1 2 4 8 and 3 5 6 7 9 0 The intersection of the set A and is A 1 i This is always true Fact A 1 i always The intersection of a set and its complement is always the empty set Weird Fact The empty set is disjoint from every set including itself Think about it Fact For A f U and B f U A 1 B f U always To watch an animation about these operations click here De Morgan s Properties These are stated as 1 and 2 Describing these properties in words 1 The complement of the union of two sets is equivalent to the intersection of the complements of the two sets 2 The complement of the intersection of two sets is equivalent to the union of the complements of the two sets These properties can be useful to resolve problems about sets Verify them for yourself by shading the areas in a Venn diagram that each side of the equality describes Notice that so far we have restricted ourselves to two sets Many problems involve three sets call them A B and C This type of problem is only modestly more difficult than those with two sets On to the next lesson Arizona State University Department of Mathematics and Statistics 3 of 3