Mathematics
American University Foundations of Mathematics Questions

American University

Question Description

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

I will provide the book we needed and the question for this work, plz explain completely and don't use mathematical symbols instead, this work only in TeX form no hand writing.

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

5.3
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
π‘‡β„Žπ‘’π...

Carnegie Mellon University

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