Set Theory & Truth Table

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Logic and Venn diagrams Set Theory Set notation Symbol Definition U Universal set Is an element or member of Is not an element of ‘ Complementary set Null Intersection Union Setset n(X) Venn diagrams IB Studies Revision Means… All the elements given in a question Number/object is in the set Number/object is not in the set Opposite of the set or ‘not the set’ An empty set with no members When an element is in two (or more sets at the same time) All the objects of two or more sets. A smaller set
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  Logic and Venn diagrams IB Studies Revision ©www.ibmaths.com  Set Theory Set notation Symbol Definition Means… U  Universal set All the elements given in a question   Is an element ormember of Number/object is in the set   Is not an element of Number/object is not in the set ‘ Complementary set Opposite of the set or ‘not the set’    Null An empty set with no members   Intersection When an element is in two (or more sets at thesame time)   Union All the objects of two or more sets.   Setset A smaller set contained wholly within a largersetn( X  ) The number of elements in a set. Venn diagrams Venn diagrams are a visual way of representing the set notation. Below are someVenn diagrams to show the set notation symbols above.A  B  Logic and Venn diagrams IB Studies Revision ©www.ibmaths.com  A  B(A  B)’ (A  B)’   Logic and Venn diagrams IB Studies Revision ©www.ibmaths.com  A’   BA’   B  Logic and Venn diagrams IB Studies Revision ©www.ibmaths.com  Guided example A group of 40 IB students were surveyed about the languages they havechosen at IB: E = English, F = French, S = Spanish.3 students did not study any of the languages above.2 students study all three languages8 study English and French10 study English and Spanish6 study French and Spanish13 students study French28 students study English(a)   Draw a Venn diagram to illustrate the data above. On your diagram writethe number in each set.(b)   How many students study only Spanish?(c)   On your diagram shade (E  F)’ , the students who do not study Englishor French.Answer (a) ã   There are 3 sets within a universal set. So the Venn diagram will have 3intersecting sets. We know that the sets must intersect as there are somestudents who belong to all 3 sets. Draw a Venn diagram with 3intersecting sets.   ã   Now we can work carefully through the statements and fill in as we goalong. ã   The 3 students who do not study any language go inside the box, butoutside of the circles. ã   2 students must go in the intersection of all three circles. ã   8 must go in the intersection of E and F, but remember that 2 havealready been filled in the intersection of all 3 sets. So only 6 will be put inthe intersection. ã   Likewise the English and Spanish intersection will have 10 – 2 = 8 and   only 4 will go in the French and Spanish intersection. ã   In the French set there should be 13, but thus far we have filled in 6+2+4= 12. So only 1 student is left to go in the French only set. ã   Likewise 28 students study English, but we have filled in 16 of thesestudents, so 12 go in the English only section. ã   There are 40 students altogether in the survey. Counting all the numbersfilled in thus far reveals we have put 36 numbers on the diagram. It isimportant to remember the 3 outside the circles, but in the box. Fill in 4 inthe Spanish only section. ã   The final diagram will look like this:
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