gen-state-the-source-coding-theorem-precisely-and-explain-why-it-f87e

State the source coding theorem precisely and explain why it matters in 2-3 sentences.

# Source Coding Theorem **Precise Statement:** For a discrete memoryless source with entropy H(X), messages of length n can be compressed to at least nH(X) bits on average with asymptotically negligible probability of information loss as n → ∞, and no compression scheme can reliably compress below this limit. **Why it matters:** The theorem establishes the fundamental limit of lossless data compression—you cannot compress data below its entropy without losing information. This has profound practical implications, explaining why we can compress text files significantly but random data hardly at all, and providing the theoretical foundation for all compression algorithms (ZIP, JPEG, MP3, etc.).