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2.1 Sets and Whole Numbers Remember to Silence Your Cell Phone and Put It In Your Bag! Sets Intuitive definition of Set – A collection of objects List the objects of a set within braces { } When listing the objects of a set, separate them with commas Use a capital letter to name a set Order of the elements is not important Element x A, x A A versus a One-to-One Correspondence Sets A and B have one-to-one correspondence iff each element of A can be paired with exactly one element of B and each element of B can be paired with exactly one element of A. Equal and Equivalent Sets Sets A and B are equal sets, symbolized by A = B, iff each element of A is also an element of B and each element of B is also an element of A. Sets A and B are equivalent sets, symbolized by A B, iff there is a one-to-one correspondence between A and B. Subset and Proper Subset For all sets A and B, A is a subset of B, symbolized as A B, iff each element of A is also an element of B. For all sets A and B, A is a proper subset of B, symbolized by A B, iff A is a subset of B and there is at least one element of B that is not an element of A. Additional Terminology The Universal Set U Empty Set or Null Set Finite Set Infinite Set { } The Complement of a Set The complement of set A, written Ᾱ, consists of all of the elements in U that are not in A. Whole Number A whole number is the unique characteristic embodied in each finite set and all the sets equivalent to it. The number of elements in set A is expressed as n(A) and is called the cardinality of set A. Counting Counting is the process that enables people systematically to associate a whole number with a set of objects. Ordering Whole Numbers For whole numbers a and b and sets A and B, where n(A) = a and n(B) = b, a is less than b, symbolized as a < b, iff A is equivalent to a proper subset of B. a is greater than b, written a > b, iff b<a Special Subsets of the Set of Whole Numbers Counting Numbers or Natural Numbers {1, 2, 3, 4, . . . } Whole Numbers {0, 1, 2, 3, . . . } Even Whole Numbers {0, 2, 4, 6, . . . } Odd Whole Numbers {1, 3, 5, 7, . . . } Three Types of Numbers Nominal Number Ordinal Number Cardinal Number An announcer says: "The student with ID number 50768-973 has just won second prize-four tickets to the big game this Saturday."