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MAT 120

MAT 120. Math for Liberal Arts. Sets and Elements. Sets are named by capital letters and indicated by set braces, { }. Elements of the set are separated by commas. A={1, 3, 6, a, b, d} 3 A but 7 A The elements may be in any order in the braces.

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MAT 120

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  1. MAT 120 Math for Liberal Arts

  2. Sets and Elements • Sets are named by capital letters and indicated by set braces, { }. • Elements of the set are separated by commas. • A={1, 3, 6, a, b, d} • 3 A but 7 A • The elements may be in any order in the braces. • Repetition of elements are not counted.

  3. Set Notations • Roster or tabular notation • Elements listed in braces. {a,b,c,d} • Use of ellipsis in roster notation. (a,b,c,…, h} • Elements must be ordered. • Need the first three elements, at least, to set pattern. • Set Builder notation • (x:x is a day of the week.} is read “The set of all elements such that the elements is a day of the week.

  4. Equal vs. Equivalent Sets • The cardinal number of a set, n(A), is the number of distinct elements in the set. • Two sets are equal if they contain the same elements. • Two sets are equivalent if they have the same number of elements. That is if n(A) = n(B).

  5. Universal Sets • A universal set is any set that contains all of the elements of all of the sets under discussion. • A = {x: x is a counting number between 4 and 7} • B = {x: xis a counting number less than 20} • C = {31, 44, 57} • A possible universal set – {1, 2, 3, …, 60} • A set that is not a universal set - • {x: x is a counting number less than 57} • A universal is called U

  6. The Empty Set • The empty set is a set that has no elements. • The empty set is sometimes called the null set. • The cardinal number of the empty set is 0. • There are two notations for the empty set. • One notation is { }. • The other is Ø. • Be careful! S = {Ø} is not an empty set because n(S) ≠ 0.

  7. Well-Defined Sets • A set is well defined if and only if it is possible to determine , without any question, whether a given element belongs to a set. • Be careful of opinions or judgments. • {x: x is a person whose height is 6 feet or more} is a well defined set.

  8. Finite vs. Infinite Sets • A set is finite if its elements can be counted. • A set is finite if it has a last element. • A set is infinite if its elements cannot be counted. • There will be another more sophisticated definition of an infinite set later.

  9. Numerals: Cardinal or Ordinal • Numerals are symbols for numbers. Each number can be represented by many numerals. • A cardinal numeral answers the question “how many.” • There are 3 books on the shelf. • An ordinal numeral answers the question “which one.” • The party starts at 5 o’clock.

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