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Let $H(X)$ denote the entropy of a discrete random variable $X$ taking $K$ possible distinct real values. Which of the following statements is/are necessarily true?

  1. $H(X) \leq \log_{2} K \; \text{bits}$
  2. $H(X) \leq H (2X)$
  3. $H(X) \leq H (X^{2})$
  4. $H(X) \leq H (2^{X})$
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