Venn Diagrams and Set Operations

Transcription

1 Survey of Math - MT 140 Page: 1 Venn Diagrams and Set Operations 1 Operations on Sets 1.1 Compliment Compliment of a Set - all the elements in the niverse that are NOT in the given set. Notation 0 in English "'s Compliment" Example 1 Given that D fxnjx < 11g and D fx 2 Njx < 10 and x is odd} Find 0 and write in Roster Form D f1; 3; 5; 7; 9g so all the elements that are not in but are still in the niverse will be: 0 D f2; 4; 6; 8; 10g Example 2 Given D fx 2 f amiliesjx are families with at least one girlg nd 0 (the compliment of Set ) We need to determine what types of elements are not in the above set. What sort of families Do Not Have "at least one girl". So this must be the families that have only boys. Thus we can write: 0 D fx 2 f amiliesjx are families with only boysg 1.2 Intersection The Intersection of Two Sets will be the elements that the two sets have in common. Symbolically this is written as \ which represents : " intersect " or "all the elements that and have in Common" Example 3 Given that: D f1; 2; 5; 7; 9; 11; 15; 17g and D f2; 4; 9; 17; 22g Find \ : To nd this all we need to do is list down what and share in common. I nd it easiest to go through each element of the rst set and ask if that element is in the other. If it is I write it down as part of my answer. Example: Is 1 in? no Is 2 in? yes, write it down and eventually arrive at the solution: \ D f2; 9; 17g

2 Survey of Math - MT 140 Page: nion nion-the combination of all the elements of two sets. Symbolical written as [ Example 4 Given D f1; 4; 7; 10g and D f2; 4; 6; 8; 10g nd [ English " union " or "list all elements of and all the elements of " This becomes {1,4,7,10,...} from the Set and then we include all the elements of that are not already listed, which in order will be {1,2,4,6,7,8,10}. *Note that I did not repeat the numbers that were in common, they only need to be listed once. 2 Venn Diagrams Venn Diagrams are pictorial representation of Sets. The General set up of a Two Set environment is depicted below. This represents all the Possibilities between two Sets and. If D, then Sector II is the only one with elements If and are DISJOINT then sector II is empty, and I and III have elements we can say that. If, then sectors I and II are the only sectors with elements Note: Sector is the place where elements that do not belong to either or are put. Here is where knowing the niverse becomes important. To construct a Venn Diagram it is usually wise to start in Sector II (the middle) and work your way out. Example 5 Construct a Venn Diagram for the given sets. D fx 2 Njx 10g D fx 2 Njx is oddg D fx 2 Njx is a prime numberg Start with what and have in common D f1; 3; 5; 7; 9g D f2; 3; 5; 7g

3 Survey of Math - MT 140 Page: 3 so: \ D f3; 5; 7g this will go in Sector II. Next ll in Sector I what hasn't been used yet in f1; 9g and what hasn't been used for in Sector III f2g Numbers that are used in neither of the Sets will be put in Sector. f4; 6; 8; 10g and we get: 1, , 6, 8, 10 Now to use this picture to answer some question becomes easy. Diagrams. I will teach this using the Sector Method of Venn Example 6 sing above data, nd: 1. \ 2. 0 \ 3. \ \ / 0 nswers: sing the sector notation we get that: 1. \ is sector II f3; 5; 7g 2. 0 D I I I; I V and D I I; I I I so what they share in common is I I I which has f2g 3. D I; I I and 0 D I; I V so what they share in common is I which has f1; 9g 4. \ is sector II so. \ / 0 will be I; I I I; I V f1; 9; 2; 4; 6; 8; 10g in order f1; 2; 4; 6; 8; 9; 10g Example 7 ll together now, lets design a Venn Diagram Set and then nswer the following questions based on the Venn Diagram D fx 2 Njx < 16g D f2; 3; 5; 7; 11; 13g D f1; 3; 6; 7; 9; 10; 13; 15g

4 Survey of Math - MT 140 Page: 4 1) Start with what elements Set and Set have in common: \ D f3; 7; 13g and put this in Sector II 2) Then what is left for the rest of f2; 5; 11g in Sector I (only write each element only ONCE for the whole Venn Diagram!!!!!!) 3) and in the rest of f1; 6; 9; 10; 15g into Sector III 4) What ever is left for the niverse, that is not already in or will go into Sector. We have: f4; 8; 12; 14g 2, 5, , 6, 9 10, 15 4, 8, 12, 14 Now lets nd: 1. [ 2. 0 [ 0 3. \ [ / 0 nswers: We will again use the Sector Method from before: 1. [ D I; I I; I I I which have the elements {2,5,11,3,7,13,1,6,9,10,15} in order f1; 2; 3; 5; 6; 7; 9; 10; 11; 13; 15g 2. 0 D I I I; I V and 0 D I; I V so 0 [ 0 D I; I I I; I V {2,5,11,1,6,9,10,15,4,8,12,14} in order f1; 2; 4; 5; 6; 8; 9; 10; 11; 12; 14; 1 3. \ 0 D I is the only sector that they share in common f2; 5; 11g 4.. [ / 0 D.I; I I; I I I / 0 D I V so...f4; 8; 12; 14g

5 Survey of Math - MT 140 Page: 5 3 Number of Elements in a nion of Two Sets If n./ D 5 and n./ D 4; then does that mean that n. [ / D 9? Remember that n./ is the number of elements in a set. Let D f1; 2; 3; 4; 5g D f4; 5; 6; 7g then [ D f1; 2; 3; 4; 5; 6; 7g so n. [ / D 7 NOT 9 So what happened? Well two of the elements where shared by both of the sets, thus if we just added the number of elements in and the number of elements in we would have counted f4; 5g twice. Formula: n. [ / D n. / C n./ n. \ / (takes away the number they have in common)