American University Foundations of Mathematics Questions

American University

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Need help with my Mathematics question - I’m studying for my class.

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American University Foundations of Mathematics Questions

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𝐴𝑐 = π‘‘β„Žπ‘’ π‘π‘œπ‘šπ‘π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘œπ‘“ 𝐴 = π‘€β„Žπ‘Žπ‘‘ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘¦π‘œπ‘’ π‘π‘Žπ‘›π‘›π‘œπ‘‘ 𝑓𝑖𝑛𝑑 𝑖𝑛 𝐴
π‘₯ ∈ 𝐴 = π‘₯ π‘π‘’π‘™π‘œπ‘›π‘”π‘  π‘‘π‘œ 𝐴 = π‘€π‘–π‘‘β„Žπ‘–π‘› π‘‘β„Žπ‘’ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘œπ‘“ 𝑠𝑒𝑑 𝐴, π‘¦π‘œπ‘’ π‘π‘Žπ‘› 𝑓𝑖𝑛𝑑 π‘₯
π‘₯ βˆ‰ 𝐴 = π‘₯ π‘‘π‘œπ‘’π‘ π‘›β€² 𝑑 π‘π‘’π‘™π‘œπ‘›π‘” π‘‘π‘œ 𝐴 = π‘€π‘–π‘‘β„Žπ‘–π‘› π‘‘β„Žπ‘’ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘œπ‘“ 𝐴, π‘¦π‘œπ‘’ π‘π‘Žπ‘›β€² 𝑑 𝑓𝑖𝑛𝑑 π‘₯
βˆ… = 𝑛𝑒𝑙𝑙 𝑠𝑒𝑑 = π‘‘β„Žπ‘’ 𝑠𝑒𝑑 π‘€π‘–π‘‘β„Ž π‘›π‘œ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  (π‘‘β„Žπ‘’ π‘’π‘žπ‘’π‘–π‘£π‘Žπ‘™π‘’π‘›π‘‘ 0 π‘“π‘œπ‘Ÿ 𝑠𝑒𝑑𝑠)
𝐴 ∩ 𝐡 = 𝐴 π‘–π‘›π‘‘π‘’π‘Ÿπ‘ π‘’π‘π‘‘π‘’π‘‘ π‘€π‘–π‘‘β„Ž 𝐡 = π‘‘β„Žπ‘’ π‘π‘œπ‘šπ‘šπ‘œπ‘› π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  𝑏𝑒𝑑𝑀𝑒𝑒𝑛 𝐴 π‘Žπ‘›π‘‘ 𝐡
𝐴 βˆͺ 𝐡 = π‘‘β„Žπ‘’ π‘’π‘›π‘–π‘œπ‘› π‘œπ‘“ 𝐴 π‘Žπ‘›π‘‘ 𝐡 = π‘Žπ‘™π‘™ π‘‘β„Žπ‘’ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘“π‘Ÿπ‘œπ‘š 𝐴 π‘Žπ‘›π‘‘ 𝐡
𝐴 βˆ’ 𝐡 = π‘‘β„Žπ‘’ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘€β„Žπ‘–π‘β„Ž π‘π‘Žπ‘› 𝑏𝑒 π‘“π‘œπ‘’π‘›π‘‘ 𝑖𝑛 𝐴, 𝑏𝑒𝑑 π‘›π‘œπ‘‘ 𝑖𝑛 𝐡
𝐴 βŠ† 𝐡 π‘šπ‘’π‘Žπ‘›π‘  𝐴 𝑖𝑠 π‘Ž 𝑠𝑒𝑏𝑠𝑒𝑑 π‘œπ‘“ 𝐡, π‘ π‘œ π‘Žπ‘™π‘™ π‘‘β„Žπ‘’ π‘’π‘šπ‘’π‘›π‘‘π‘  π‘œπ‘“ 𝐴 π‘Žπ‘Ÿπ‘’ 𝑖𝑛 𝐡 π‘Žπ‘  𝑀𝑒𝑙𝑙
𝐴 βŠ‡ 𝐡 = 𝐡 𝑖𝑠 π‘Ž 𝑠𝑒𝑏𝑠𝑒𝑑 π‘œπ‘“ 𝐴, π‘ π‘œ π‘Žπ‘™π‘™ π‘‘β„Žπ‘’ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘π‘  π‘œπ‘“ 𝐡 π‘Žπ‘Ÿπ‘’ 𝑖𝑛 𝐴 π‘Žπ‘  𝑀𝑒𝑙𝑙
𝐴 Γ— 𝐡 = π‘π‘Žπ‘Ÿπ‘‘π‘’π‘ π‘–π‘Žπ‘› π‘π‘Ÿπ‘œπ‘‘π‘’π‘π‘‘ π‘œπ‘“ 𝐴 π‘Žπ‘›π‘‘ 𝐡
= π‘Ž 𝑠𝑒𝑑 π‘œπ‘“ π‘œπ‘Ÿπ‘‘π‘’π‘Ÿπ‘’π‘‘ π‘π‘Žπ‘–π‘Ÿπ‘  π‘€π‘–π‘‘β„Ž π‘‘β„Žπ‘’ π‘“π‘–π‘Ÿπ‘ π‘‘ π‘’π‘™π‘’π‘šπ‘’π‘›π‘‘ π‘“π‘Ÿπ‘œπ‘š 𝐴 π‘Žπ‘›π‘‘ π‘‘β„Žπ‘’ π‘ π‘’π‘π‘œπ‘›π‘‘ π‘“π‘Ÿπ‘œπ‘š 𝐡

1. a) (𝐴𝑐 )𝑐 = 𝐴
π‘₯ ∈ (𝐴𝑐 )𝑐 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 π‘₯ βˆ‰ 𝐴𝑐
π‘₯ βˆ‰ 𝐴𝑐 𝑖𝑓 π‘Žπ‘›π‘‘ π‘œπ‘›π‘™π‘¦ 𝑖𝑓 π‘₯ ∈ 𝐴

Carnegie Mellon University

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