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10/9/18 Working with Sets Algebra Section 3.5 Do you remember these vocabulary words? If not, go back through your notes or textbook to refresh your memory. Real numbers Rational numbers Whole number Integers Prime Composite 1 10/9/18 Introduction A set is a collection of objects. Example: { -4, -3, -2, -1, 0, 1, 2, 3, 4} A subset is a subgroup of a set. – A subset of the above set could be all even #s { -4, -2, 0, 2, 4} – A subset of the above set could be all positive #s { 1, 2, 3, 4} The objects (each number) in a set are called elements of the set. Writing a Set When writing a set of numbers, a set must: 1. be written using braces { }. 2. be labelled with a capital letter. 3. be written with the values listed from least to greatest from left to right (just like on a number line). 4. not have duplicate elements. For example the following is labelled as “Set Z”: Z = { …-3, -2, -1, 0, 1, 2, 3, …} 2 10/9/18 Two Ways to Write a Set There are two ways to write a set of numbers; Roster form or Set-builder Notation Roster Form Roster form is the method of describing a set by listing each element of the set. Example: Let M = The set of all 8th grade math teachers at Martino. Roster form is: M = {Bahret, Lavey, Proff, Sterritt} (Since there are only 4 names it can clearly be listed as a set.) 3 10/9/18 Roster Form Another example… Let set A = The set of odd numbers greater than zero, and less than 10. Roster form is: A = {1, 3, 5, 7, 9} (**Make sure all number appear in order from least to greatest.) Roster Form Sometimes we can’t list all the elements of a set. For instance, Z = The set of all integer numbers. We can’t write out all the integers because there are an infinite number of integers, so we then use dots. The dots mean continue on in this pattern forever and ever. Z = { …-3, -2, -1, 0, 1, 2, 3, …} This is the set of whole numbers. W = {0, 1, 2, 3, …} 4 10/9/18 Set – Builder Notation When it is not convenient or not possible to list all the elements of a set, we use a notation called set – builder notation. Set-builder notation is basically a written description of the elements in a set. If possible, use an inequality sign as part of the description. Set – Builder Notation Example: Let set A = The set of all real numbers greater than 5. Set-builder notation is: A = {x | x > 5} This set is read as, “all elements x such that x is all real numbers greater than 5”. 5 10/9/18 Set – Builder Notation Another Example: Let set D = The set of all prime numbers greater than 2. Set-builder notation is: D = {x | x is prime and x > 2} This set is read as, “all elements x such that x is prime and x is greater than 2”. Roster form vs. Set-builder Notation J = The set of all whole numbers greater. than 5. Roster Form: J = {6, 7, 8, 9,…} Set-builder Notation: J = {x | x is all whole number where x > 5}. 6 10/9/18 Roster form vs. Set-builder Notation N = The set of natural numbers. Roster Form: N = {1, 2, 3, …} Set-builder Notation: N = {x | x is all natural numbers where, x > 0}. You can write this set in set-builder notation, however, is a lot longer to write---it is more convenient to write the set in roster form. Roster form vs. Set-builder Notation Q = The set of rational numbers Roster Form: Cannot write in roster form, as you cannot list all rational numbers. The list would go on forever. Set-builder Notation: Q ={x | x are all rational numbers} 7 10/9/18 Review Problem 1 & Got it 1 on pg. 194. (Answers to the Got its are in the back of the textbook.) Review Problem 2 & Got it 2 on pg. 195. (Answers to the Got its are in the back of the textbook.) 8 10/9/18 Empty Set An empty set is a special set. It contains no elements. (It can also be called a null set.) It is usually written as { } or Æ Do not be confused by this question: Is the set {0} empty? Of course, not! It is not empty! It contains the element zero. Universal Set The Universal Set (U ): • is the set of all possible elements used in a problem. • can also be called the universe. • is always the largest set you are working with. For the set of the students in this class, U would be all the students at Martino (or perhaps all the students in the world). For the set of the vowels of the alphabet, U would be all the letters of the alphabet. 9 10/9/18 Universal Set and Subsets When every element of one set is also an element of another set, we say the first set is a subset. Example A= {1, 2, 3, 4, 5} and B ={2, 3} We say that B is a subset of A. The symbol for subset is Í. We can then say, B ÍA. Universal Set and Subsets Another example: Let S= {1,2,3}, list all the subsets of S. The subsets of S are Æ , {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}. **The empty set is always considered a subset of any set. 10 10/9/18 Review Problem 3 & Got it 3 on pg. 195. (Answers to the Got its are in the back of the textbook.) Complement of a Set The complement of a set is the set of all elements in the universal set that are NOT in the set. The complement of Set A would be written as A’. The complement of set S would be written as S’ . 11 10/9/18 Venn Diagrams Venn Diagrams represents sets graphically – The box represents the universal set, U – Circles represent the set(s), S S’ U S – Inside the box, but outside the circle represents the complement of the set, S’ Venn Diagram Example Consider set S, which is the set of all vowels in the alphabet. Remember: – The circle represents the set (all vowels) – The box represents the b c d f S’ universal set (all letters in g h j S the alphabet) k l m – Inside the box, but outside n p q e a the circle represents the r s t o complement of the set v w x (all consonants) y z U i u 12 10/9/18 Venn Diagram Example The Venn Diagram can look a little too busy, so usually the elements within the complement of the set are not written out. So the final Venn Diagram would look like this. U S’ S e a o i u Review Problem 4 & Got it 4 on pg. 196. (Answers to the Got its are in the back of the textbook.) 13 10/9/18 Now you should be ready for the homework. But keep this PPT at your finger tips, in case you need to reference it. 14