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 9 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
Sets as a Basis for Whole Numbers Revised 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 m a t h e i c 5 How many members does it has nS 8 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 e 1 the set of all US states 0 S 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 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 m3 a Two sets are disjoint if they have no elements in common 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 lfthe universe is the set of the 26 letters of our alphabet the complement of the set of the letters in the word mathematics is the set of the letters of the alphabet not in the word mathematics TOD O 5c rlt 551 Meme Cartesian Product of a Set The Cartesian Product named after Rene Descartes of two setsA and B is the set of ordered pairs which results from matching every element of A with every element of B Example IfA m a t h andB c l a 5 write the Cartesian Product ofA and B Exercises SupposeA 0 1 B 1 2 3 C 0 1 2 Sketch a Venn Diagram to illustrate their relationship Here is one possible Venn diagram A B C We can resize the circles to show thatA is entirely contained in C and that 2 is a member of both B and C but not inA and 3 is a member only to B So this Venn Diagram is also correct B C Insert the appropriate symbol 6 g C or g in the blank to make a true statement 2 B A B C A U C 2 C A C A A m C 2 e B 2 as C A g B A is not a subset ofB since 0 is inA but 0 is not a member ofB A C C A is a proper subset ofC since all ofA is contained in C and 2 is a member ofC that is not in A C gAUC sinceAUC 0l2 andC0l2 A gA C sinceA C 0landA0l Closure Property for Addition A set of numbers is closed under addition if the sum of any two members including themselves is also in that set A 0 1 B 1 23 C0 12 Which of the sets A B and C are closed under addition NotA ll2 and 2 A Not B 235 and 5 B Not C 123 and 3 C Similarly which are closed under subtraction NotA Oill and 1 A NotB 220 and 0 B Not C 022 and 2 C Which of the sets are closed under multiplication Just set A NotB orC2gtlt 24 and4 B and4 C Which of the sets are closed under division NotA or C l 0 is undefined and in neither set NotB 12 2 and 2 B
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