Chapter 2: Q38E (page 137)
Show that if A and B are sets, then
a) \(A \oplus B{\bf{ = }}B \oplus A\)
b) \(\left( {A \oplus B} \right) \oplus B = A\)
(a) \(A \oplus B = B \oplus A\)
(b) \((A \oplus B) \oplus B = A\)
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Determine whether each of these functions from Z to Z is one-to-one.
Question: Let and let for all . Show that f(x) is strictly increasing if and only if the function is strictly decreasing.
a) define the floor and ceiling functions from the set of real numbers to the set of integers.
b) For which real numbersis it true that
a) Show that if a set S has cardinality m, where m is a positive integer, then there is a one-to-one correspondence between S and the set {1,2,...m}
b) Show that if S and T are two sets each with m elements, where m is a positive integer, then there is a one-to-one correspondence between S and T.
For each of these partial functions, determine its domain, codomain, domain of definition, and the set of values for which it is undefined. Also, determine whether it is a total function.
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