Problem 34E

Show that (0. 1) and R have the same cardinality. [Hint: Use the Schroder-Bernstein theorem.]

Survey of Math: Week Two Sets Set: A list of elements separated by commas. Order of elements does not matter. Multiplicity does not matter o Ex: {a, a, a, b, c, c} is the same as {a, b, c} The elements are always contained in {} brackets. This is how you know if you are dealing with a set. Two sets are equal if they have the exact same elements in them. Ex: {a, b, c} = {c, b, a} Cardinality of sets: The number of distinct elements in that set. Ex: If set A= {a, 1, b, 2, c, 3}, then the cardinality of A is six, or │A│= 6, or n│A│= 6. Cardinality can be expressed as │x│= c or n│x│= c where x is the label for the set and c is the cardinality. The empty set: A set with a cardinality of zero. It is expressed as ø. A specific element in a set is expressed as Є. Ex: If A={1, 2, 3}, then 2Є│A│. A subset is expressed as C. Set A is a subset of Set B if all of the elements of A can be found in B. Ex: If A= {a, b} and B= {a, b, c, d}, then A C B. The empty set is always considered a subset. A proper subset is a subset which is not equal to the original set. If B= {a, b, c, d} and D={a, b, c, d}, then D C C, but it is not a proper subset because they are equal. If E={a, b, c}, then E is a proper subset of both B and D because E is not equal to them. The universal set is the set which contains all elements. Every possible set is a subset of the universal set. Well-Defined vs. Not Well-Defined Sets