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# Chapter 2. Section 1 MA 201

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This 3 page Class Notes was uploaded by Keturah Hrebicik on Friday February 12, 2016. The Class Notes belongs to MA 201 at University of Kentucky taught by Staff in Winter 2016. Since its upload, it has received 19 views. For similar materials see in Mathematics (M) at University of Kentucky.

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Date Created: 02/12/16

MA 201 Chapter 2. Section 1. Sets: a collection of objects Elements of a set are the same thing as the objects that make up the set. Sets are normally denoted add capital letters. Set notation: List notation: C={1,2,3,4,5}, where C is the name of the set. Set builder notation: C={|x|x satisfies some condition}. For example, if the condition states that x is an even number between 1 and 10 then the set would look like this in list notation: {2,4,6,8} Set equality Sets are simply equal if they contain the same elements. For example: {2,3,5} = {|x| x is a prime number less than 6} Null set (empty set) A null set is a set with no elements Examples: find the elements of the following sets A={x|x is an odd number between zero and ten} {1,3,5,7,9} B={x| x is an even number between one and thirteen) {2,4,6,8,10,12} C=0 {} D={x|x is an even number that is less than 2,} {} Cardinality of a set The number of elements in a set (this can either be finite or infinite) |x| is the way top denote the cardinality of set A Natural numbers are counting numbers. Starting at one. Whole numbers are counting numbers but include zero. Equivalent sets Sets are deemed equivalent if they have the same cardinality To denote equivalent sets you use a ~. {1,2,3}~{cat, dog, penguin} Subsets Set A is a subset of set B if every element in set A is also in set B. We show this by writing A C B Examples: {1,2,3} C {1,2,3,4}, {2,6,5} C {1,2,3,4,5,6}, {x| x is an even numbers} C {1,2,3,4....} Non examples: {A,1,2} C {1,2,3} Note: if A is any set, the null set of A is a sub set. Note: Any set is a set of itself. Example: {1,2,3} C {1,2,3} We say set A is a proper subset of set B, if A C B and there is some element of B which is not in A. This means the set are not equal. We write A C B to show A is a proper subset of B. List all of the subset of the following: How many subsets are there? Is there a pattern? I. {A}: There are two subsets. {}, {A} II. {A,B}: There are 4 subsets. {A}, {B}, {A,B}, {} III. {A,B,C}: There are 8 subsets. {A}, {B}, {C}, {A,B}, {B,C}, {A,C}, {A,B,C}, {} Generally in a set with N elements, the amount of subsets can be determined by 2n Subsets and cardinality Given finite sets A and B, if A C B then |A| < |B| If A C B then | A| < |B| Example: A= {A,B,C} B={1,2,3,4} A C B and |A| < |B| 3<4 Recall a one to one correspondence between sets A and B is a pairing elements. Both A and B have the same amount of elements. {1,2,3, 4} Both of these sets have a one to one correspondence and have the {A,B,C,D} same cardinality Infinite sets and cardinality For infinite sets, they have the same size/cardinality if they have a one to one cor respondence. All infinity are the same size. If there is two sets that are infinite in size then they have the same cardinality.

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