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Discrete Mathematics Exercises

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Problem 1 If 𝐴, 𝐡 and 𝐷 are three events such that π‘ƒαˆΊπ΄ βˆͺ 𝐡 βˆͺ 𝐷ሻ = 0.7, what is the value of π‘ƒαˆΊπ΄πΆ ∩ 𝐡𝐢 ∩ 𝐷 𝐢 ሻ? 𝐴𝐢 = π‘ˆ βˆ’ 𝐴 Given we know { 𝐡𝐢 = π‘ˆ βˆ’ 𝐡 , 𝑀𝑒 π‘π‘Žπ‘› π‘‘π‘Ÿπ‘Žπ‘€ π‘‘β„Žπ‘’ π‘‘π‘–π‘Žπ‘”π‘Ÿπ‘Žπ‘šπ‘  π‘Žπ‘  π‘‘β„Žπ‘’ π‘“π‘œπ‘™π‘™π‘œπ‘€π‘–π‘›π‘”: 𝐷𝐢 = π‘ˆ βˆ’ 𝐷 a) 𝐴 βˆͺ 𝐡 βˆͺ 𝐷 b) 𝐴𝐢 ∩ 𝐡𝐢 ∩ 𝐷 𝐢 We can see π‘ƒαˆΊπ΄πΆ ∩ 𝐡𝐢 ∩ 𝐷 𝐢 ሻ = π‘ƒαˆΊπ΄ βˆͺ 𝐡 βˆͺ 𝐷ሻ𝐢 Therefore, π‘ƒαˆΊπ΄πΆ ∩ 𝐡𝐢 ∩ 𝐷 𝐢 ሻ = 1 βˆ’ 0.7 = 0.3 Problem 2 Prove the following identity for any three events 𝐴, 𝐡, π‘Žπ‘›π‘‘ 𝐢: π‘ƒαˆΊπ΄ βˆͺ 𝐡 βˆͺ 𝐢ሻ = π‘ƒαˆΊπ΄αˆ» + π‘ƒαˆΊπ΅αˆ» + π‘ƒαˆΊπΆαˆ» βˆ’ π‘ƒαˆΊπ΄ ∩ 𝐡ሻ βˆ’ π‘ƒαˆΊπ΄ ∩ 𝐢ሻ βˆ’ π‘ƒαˆΊπ΅ ∩ 𝐢ሻ + π‘ƒαˆΊπ΄ ∩ 𝐡 ∩ 𝐢ሻ We can rewrite π‘ƒαˆΊπ΄ βˆͺ 𝐡 βˆͺ 𝐢ሻ π‘Žπ‘  𝑃(ሺ𝐴 βˆͺ 𝐡ሻ βˆͺ 𝐢). And we know for any two events 𝑋 π‘Žπ‘›π‘‘ π‘Œ: π‘ƒαˆΊπ‘‹ βˆͺ π‘Œαˆ» = π‘ƒαˆΊπ‘‹αˆ» + π‘ƒαˆΊπ‘Œαˆ» βˆ’ π‘ƒαˆΊπ‘‹ ∩ π‘Œαˆ» Using that, we can write: ...
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