Define mappings alpha and beta from the set of natural numbers, N, to itself by

Question

Define mappings alpha and beta from the set of natural numbers, N, to itself by alpha(n) = 2n and beta(n) = (n+1)/2 if n is odd and n/2 if n is even. Then alpha is one-to-one but not onto; and beta is onto but not one-to-one. Now, for each n?N, the equation beta(x) = n has exactly two solutions. The solutions of beta(x) = 2 are x = 3 and x = 4, for example).

a) Define a mapping y : N --> N such that for each n?N the equation y(x) = n has exactly three solutions.

b) Define a mapping y : N --> N such that for each n?N the equation y(x) = n has exactly n solutions. (It suffices to describe y in words).

c) Define a mapping y : N --> N such that for each n?N the equation y(x) = n has infinitely many solutions.

Explanation / Answer

we see that f is a one-to-one map of S onto S ... set of natural numbers N, (2) the set K = {1/n| n ... N+1} a_{n-N-1} + ... + beta_n a_0| _

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