What is the complete proof for INCLUSION-EXCLUSION FORMULA and BOOLE'S INEQUALITY? Step by Step Solutions. The BOOLE'S INEQUALITY that I meant was both: 𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) ≤ 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛) or 𝑃(⋃ 𝐴𝑖 𝑛 𝑖=1 ) ≤ ∑ 𝑃(𝐴𝑖) 𝑛 𝑖=1 . 𝑃(𝐴1 ∪ 𝐴2 ∪ … ∪ 𝐴𝑛 ) ≥ 𝑃(𝐴1 ) + 𝑃(𝐴2 ) + ⋯ + 𝑃(𝐴𝑛) or 𝑃(⋃ 𝐴𝑖 𝑛 𝑖=1 ) ≥ ∑ 𝑃(𝐴𝑖) 𝑛 𝑖=1 .
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