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Part BCSIR NET December 2023

Measurable sets and functions

Let X be a non-empty finite set and Y = { : f is a real-valued function on X}. Which one of the following statements is true?

  1. A.Y is an infinite set.
  2. B.Y has elements.✓
  3. C.There is a bijective function from X to Y.
  4. D.There is a surjective function from X to Y.

Solution

Every subset of X is the zero set of its indicator-complement (f = 1 off the subset, 0 on it), so Y is the power set of X, with > |X| elements.

Related counterexample: A subset of ℝ with measure zero is countable

More on this topic

From Lebesgue Measure and Integration › Measurable sets and functions

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