Prove that a function f is 1-1 iff for all . Given that .
Prove that a function if is onto iff for all . Given that .
(a) How many functions are there from a non-empty set S into \?
(b) How many functions are there from into an arbitrary set ?
(c) Show that the notation implicitly involves the notion of a function.
Let be a function, let , , and . Prove that
Let I be a non-empty set and for each , let be a set. Prove that
(a) for any set B, we have
(b) if each is a subset of a given set S, then where the prime indicates complement.
Let A, B, C be subsets of a set S. Prove the following statements:
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