Math Elem Teachers I
Math Elem Teachers I MA 101
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This 0 page Class Notes was uploaded by Fiona Sipes on Sunday November 1, 2015. The Class Notes belongs to MA 101 at Indiana University Purdue University - Fort Wayne taught by John LaMaster in Fall. Since its upload, it has received 11 views. For similar materials see /class/233530/ma-101-indiana-university-purdue-university-fort-wayne in Mathematics (M) at Indiana University Purdue University - Fort Wayne.
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Date Created: 11/01/15
0000000000 AVAVAVAVAV 3838383838 CQCDCQCDCD DQDQDGDQDQ Sets as a Basis for Whole Numbers A set is a collection of objects and the objects are called elements or members of the set The empty set or null set written or Q is the set without any members ie the set of all US states bordering Antarctica is the empty set A nonempty set is finite if it can have its elements listed where the list eventually ends while an infinite set goes on without end ie the set ofintegers 3 2 l 0 l 2 3 is in nite The cardinality of a nite setA written nA is how many members it has What is S if the set S consists of the letters in the word mathematics S How many members does it has nS Subset of a Set A QB and Proper Subset of a Set A CB SetA is a said to be a subset ofB written A QB if and only if every element ofA is an element ofB SetA is a said to be a proper subset ofB writtenA CB ifA QB and there is an element ofB that is not inA Example The set of the original thirteen colonies is a proper subset of the set of all US states The set of letters in the word math is a proper subset of the set of letters in the word mathematics m a t h C m a t h e i c s 61 CS Union of Sets AU B The union of two setsA and B written A U3 is the set of all elements belonging to eitherA or to B or both What letters are in the word math or in the word class The union helps us answer that question IfA m a t h B c 1 a s then A U B m a t h c 1 s Intersection of Sets Am B The intersection of two setsA and B written Am B is the set of all elements common to setsA and B What letters are in the word math and in the word class The intersection helps us answer that question EA m a t h B 0 1 a 5 then f 33 a Two sets are disjoint if they have no elements in common d 0 What letters are in the word math and in the word dog Complement of a Set A The complement of a setA writtenA is the set of all elements in the universe that are not inA 111 lfthe umverse 1s the set of the 26 letters of our alphabet l 1 10 the complement of the set of the letters in the word mathematics is qr n v the set of the letters of the alphabet not in the word mathematics 39XV Z Exercises SupposeA 0 l B 1 2 3 C 0 l 2 Sketch a Venn Diagram to illustrate their relationship Insert the appropriate symbol 6 e g C or t in the blank to make a true statement 2 B A B C AUC 2 C A C A AnC Closure Property for Addition A set of numbers is closed under addition if the sum of any two elements including themselves is also in that set Which of the sets A B and C if any are closed under addition Similarly which of the sets A B and C if any are closed under subtraction Which of the sets A B and C if any are closed under multiplication Which ofthe sets A B and C if any are closed under division A Better Numera on System After some time there was a conference called so that the mathematicians could get together to share their ideas Excitement was high as the Xrnanian mathematicians began to arrive at the conference Several proposals for new number systems were on the program but the one that generated the most excitement was a system which only used the symbols quot0 quotAquot quotBquot quotCquot and quotDquot It was rumored that this system solved the quotmanyquot problem you could count as high as you wanted and you could identify exactly how many items there were in a set To everyone39s amazement it was further rumored that the mathematician had developed algorithms for adding subtracting multiplying and dividing within the new system As the day of the presentation dawned mathematicians from all over Xrnania began to ll the auditorium Soon all the seats were lled and in no time the aisles were also packed A respectful hush fell across the crowd as the presider stepped up to the podium to introduce the inventor of the new system quotIt is my sad duty to announce that our most esteemed colleague and friend the inventor of the system about which we are all so excited passed away last night She was very old and she quietly slipped away from us in her sleep She was a kind person a rst rate mathematician a respected colleague and a great teacher We will miss her deeply quotWe have not been able to locate any written record of the system she was about to share with us The only thing we have to provide us with a clue about her work is a set of blocks It is my sincere hope that we the mathematicians of Xmania will be able to work together to reconstruct the system that our dear colleague had planned to share with us Your research group has decided to take on the challenge to reconstruct the nurneration system that went the way of the famous mathematician Construct a nurneration system which meets all the characteristics of the mathematician39s system This means that the system 1 bears a clear and distinct relationship to the set ofblocks she le behind39 2 uses only the symbols 0 A B C and D39 3 allows us to count as high as we want39 and 4 allows us to uniquely identify the number of items in any collection IMPORTANT NOTE To keep from totally confusing the Xmanians you should keep the meaning of the symbols 0 A B c D the same as they39ve always hem This means that 0 represents No items A represents I B represents 0 O C represents 0 O O D represents 0 O O O s i i l 39 butyoucanrefatotheoldsysta39nto 39 whatd already 39 39 quot help strategy make a 39 n w a Include any information ieillustrations examples explanations etc that would be necessary for a ic t ov e 1 learn how to count inyour systa39n 2 L L quot quot tau 3 undersmndwhy your strategy makes smse 4 be convinced thatyour systen will allow her to use only the symbols 0 A B c and D to mquzly idmtify the number of ita39ns in any collection and t mm W n Remember u Hm MI W u w 39 39 39 39 idea that maniaquot can
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