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2.1 Basic Set Concepts Set 2 : a number of things of the same kind that belong or are used together [an electric train set] 21 : a collection of elements and especially mathematical ones (as numbers or points)- called also class (MWD) The objects in a set are called elements or members. A set must be well defined. The elements must be clearly determined. From the definition of the set we must be able to determine if an element is or is not a member of the set. The days of the week form a set. Tuesday is an element of the set of days of the week. Using the roster method, we list the elements of the set. W = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday Will the great presidents of the United States form a set? Is the group well defined? Express the set of months of the year using the roster method. Write a word description of the set L={a, b, c, d, e} The set of the first five letters of the alphabet. Set Builder Notation J is the set of all x such that x is a month with a name that starts with J Express the set W={2, 4, 6, 8} using set builder notation. Express the set E={x | x is month that begins with the letter E} using the roster method. There are no elements in set E. We express this as { } or Ø. We call a set with no elements an empty set or a null set. p is an element of the set containing n, o, p, q, r, s. k is not an element of the set containing n, o, p, q, r, s. Which symbol will make the statement true? Natural or Counting Numbers ¥ = { 1, 2, 3, 4, 5, 6, …} Set B is the set of Natural numbers less than 4. Express B using the roster method. Set R is the set of Natural numbers between 7 and 11. Express R using the roster method. Set L is the set of Natural numbers between 25 and 412. Express L using the roster method. Cardinal Number The cardinal number of a set is the number of elements in the set. If A={3, 7, 8, d, e}, then n(A)=5 What is the cardinality of M, if M={6, 7, 8, 9}? n(M)=4 What is the cardinality of P, if P={1, 2, 3, …16, 17}? n(P)=17 What is the cardinality of Z, if Z={4, 5, 6, …23, 24}? n(Z)=21 What is the cardinality of Q, if Q={2, 4, 6, … 32, 34}? n(Q)=17 What is the cardinality of A, if A={1, 3, 5, …45, 47}? n(A)=24 What is the cardinality of Y, if Y={7, 10, 13, …94, 97}? n(Y)=31 Sets are equivalent if they have the same cardinality. { Ω, π, µ, ∂} is equivalent to { 5, 7, 13, f } Is {a, b, d, e, a} equivalent to {3, 4, 5, 6} ? They both have a cardinality of 4. They are equivalent sets. Is {1, 3, 5, 3, 5} equivalent to {A, B, A, 6} ? They both have a cardinality of 3. They are equivalent sets. Set A is a finite set if n(A)=0 or n(A) is a natural number. A set that is not finite is called an infinite set. {1, 2, 4, 8, 16, 32, … } is an infinite set. Sets are said to be equal if they contain exactly the same elements. {Ralph, Alice, Ed, Trixie}={Alice, Trixie, Ed, Ralph} {x|x is a natural number less than 100}={1, 2, 3, … 98, 99} { 2, 3, 4, 5, 2, 3 } = { 5, 3, 2, 4 }