เซต

1. 𝑺𝒆𝒕𝒔
𝑶𝒗𝒆𝒓𝒗𝒊𝒆𝒘
𝑺𝒆𝒕𝒔
𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒆𝒕𝒔 𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝒔𝒆𝒕𝒔 𝑹𝒆𝒍𝒂𝒕𝒊𝒐𝒏𝒔 𝒐𝒇 𝒔𝒆𝒕𝒔
−𝑈𝑛𝑖𝑜𝑛 (∪) −𝐸𝑚𝑝𝑡𝑦 𝑠𝑒𝑡𝑠 −𝐸𝑞𝑢𝑎𝑙
− 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 (∩) − 𝐹𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡𝑠
− 𝐷𝑖𝑓𝑓𝑒𝑟𝑟𝑒𝑛𝑐𝑒 (−) − 𝐼𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡𝑠 −𝑆𝑢𝑏𝑠𝑒𝑡𝑠
− 𝐶𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 (′) − 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑠𝑒𝑡𝑠 −𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡s
𝑽𝒆𝒏𝒏 − 𝑬𝒖𝒍𝒆𝒓′𝒔 𝒅𝒊𝒂𝒈𝒓𝒂𝒎
𝑾𝒉𝒂𝒕 𝒊𝒔 𝒂 𝒔𝒆𝒕?
− 𝐴 𝒔𝒆𝒕 𝑖𝑠 𝑎 𝑐𝑜𝑙𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑜𝑏𝑗𝑒𝑐𝑡𝑠, 𝑡𝑕𝑖𝑛𝑔𝑠 𝑜𝑟 𝑠𝑦𝑚𝑏𝑜𝑙𝑠 𝑤𝑕𝑖𝑐𝑕 𝑎𝑟𝑒 𝒄𝒍𝒆𝒂𝒓𝒍𝒚 𝒅𝒆𝒇𝒊𝒏𝒆𝒅.
− 𝑇𝑕𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑜𝑏𝑗𝑒𝑐𝑡𝑠 𝑖𝑛 𝑎 𝑠𝑒𝑡 𝑎𝑟𝑒 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡𝑕𝑒 𝒎𝒆𝒎𝒃𝒆𝒓𝒔 𝑜𝑟 𝒆𝒍𝒆𝒎𝒆𝒏𝒕𝒔 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑒𝑡.
𝑆𝑒𝑡 = {𝑚𝑒𝑚𝑏𝑒𝑟1, 𝑚𝑒𝑚𝑏𝑒𝑟2, 𝑚𝑒𝑚𝑏𝑒𝑟3}
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝑆𝑒𝑡 𝑜𝑓 𝑑𝑎𝑦 = {𝑆𝑢𝑛𝑑𝑎𝑦, 𝑀𝑜𝑛𝑑𝑎𝑦, 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦, 𝑇𝑕𝑢𝑟𝑠𝑑𝑎𝑦, 𝐹𝑟𝑖𝑑𝑎𝑦, 𝑆𝑎𝑡𝑢𝑟𝑑𝑎𝑦}

2. 𝑾𝒓𝒊𝒕𝒊𝒏𝒈 𝑺𝒆𝒕𝒔
𝑇𝑕𝑒𝑟𝑒 𝑎𝑟𝑒 𝑡𝑤𝑜 𝑤𝑎𝑦𝑠 𝑡𝑜 𝑤𝑟𝑖𝑡𝑒 𝑠𝑒𝑡𝑠 ∶
1) 𝐿𝑖𝑠𝑡𝑖𝑛𝑔 𝑚𝑒𝑡𝑕𝑜𝑑 ∶ 𝐴𝑙𝑙 𝑜𝑓 𝑡𝑕𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝑎𝑟𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛, 𝑠𝑢𝑐𝑕 𝑎𝑠 𝐴 = 1,3,5,7,9 .
2) 𝑆𝑒𝑡 − 𝑏𝑢𝑖𝑙𝑑𝑒𝑟 𝑚𝑒𝑡𝑕𝑜𝑑 ∶ 𝐴 𝑡𝑦𝑝𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑛𝑎𝑚𝑒𝑑, 𝑎𝑙𝑜𝑛𝑔 𝑤𝑖𝑡𝑕 𝑖𝑡𝑠 𝑑𝑒𝑠𝑐𝑟𝑖𝑝𝑡𝑖𝑜𝑛,
𝑠𝑢𝑐𝑕 𝑎𝑠 𝐴 = {𝑥|𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟 𝑓𝑟𝑜𝑚 1 𝑡𝑜 10}.
𝑁𝑜𝑡𝑒: 𝑇𝑕𝑒 𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑏𝑎𝑟 𝑖𝑠 𝑟𝑒𝑎𝑑 "such that"
𝑴𝒆𝒎𝒃𝒆𝒓𝒔 𝒐𝒇 𝒔𝒆𝒕𝒔
𝑊𝑒 𝑟𝑒𝑙𝑎𝑡𝑒 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑎𝑛𝑑 𝑎 𝑠𝑒𝑡 𝑢𝑠𝑖𝑛𝑔 𝑡𝑕𝑒 𝑠𝑦𝑚𝑏𝑜𝑙 ∈. 𝐼𝑓 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑠𝑒𝑡 𝐴,
𝑤𝑒 𝑤𝑟𝑖𝑡𝑒 𝑥 ∈ 𝐴. 𝐼𝑓 𝑎𝑛 𝑜𝑏𝑗𝑒𝑐𝑡 𝑥 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑠𝑒𝑡 𝐴, 𝑤𝑒 𝑤𝑟𝑖𝑡𝑒 𝑥 ∉ 𝐴
∈ 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 “𝒊𝒔 𝒂𝒏 𝒆𝒍𝒆𝒎𝒆𝒏𝒕 𝒐𝒇’ 𝑜𝑟 “𝑖𝑠 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓” 𝑜𝑟 “𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜”
∉ 𝑑𝑒𝑛𝑜𝑡𝑒𝑠 “𝒊𝒔 𝒏𝒐𝒕 𝒂𝒏 𝒆𝒍𝒆𝒎𝒆𝒏𝒕 𝒐𝒇” 𝑜𝑟 “𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓” 𝑜𝑟 “𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜”
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐼𝑓 𝐴 = {1, 3, 5} 𝑡𝑕𝑒𝑛 1 ∈ 𝐴 𝑎𝑛𝑑 2 ∉ 𝐴
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔 𝒐𝒇 𝒔𝒆𝒕𝒔
𝑬𝒎𝒑𝒕𝒚 𝑺𝒆𝒕 𝒐𝒓 𝑵𝒖𝒍𝒍 𝑺𝒆𝒕
𝑇𝑕𝑒𝑟𝑒 𝑎𝑟𝑒 𝑠𝑜𝑚𝑒 𝑠𝑒𝑡𝑠 𝑡𝑕𝑎𝑡 𝑑𝑜 𝑛𝑜𝑡 𝑐𝑜𝑛𝑎𝑡𝑖𝑛 𝑎𝑛𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑎𝑡 𝑎𝑙𝑙. 𝑊𝑒 𝑐𝑎𝑙𝑙 𝑎 𝑠𝑒𝑡 𝑤𝑖𝑡𝑕 𝑛𝑜
𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑡𝑕𝑒 𝒏𝒖𝒍𝒍 𝑜𝑟 𝒆𝒎𝒑𝒕𝒚 𝑠𝑒𝑡. 𝐼𝑡 𝑖𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑒𝑑 𝑏𝑦 𝑡𝑕𝑒 𝑠𝑦𝑚𝑏𝑜𝑙 { } 𝑜𝑟 Ø .
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
- The set of months with 32 days.
- The set of squares with 5 sides.
- 𝐴 = {}
- 𝐵=∅

3. 𝑭𝒊𝒏𝒊𝒕𝒆 𝑺𝒆𝒕𝒔
𝑭𝒊𝒏𝒊𝒕𝒆 𝒔𝒆𝒕𝒔 𝑎𝑟𝑒 𝑠𝑒𝑡𝑠 𝑡𝑕𝑎𝑡 𝑕𝑎𝑣𝑒 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑚𝑒𝑚𝑏𝑒𝑟𝑠. 𝐼𝑓 𝑡𝑕𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒
𝑠𝑒𝑡 𝑎𝑟𝑒 𝑙𝑖𝑠𝑡𝑒𝑑 𝑜𝑛𝑒 𝑎𝑓𝑡𝑒𝑟 𝑎𝑛𝑜𝑡𝑕𝑒𝑟, 𝑡𝑕𝑒 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑤𝑖𝑙𝑙 𝑒𝑣𝑒𝑛𝑡𝑢𝑎𝑙𝑙𝑦 “𝑟𝑢𝑛 𝑜𝑢𝑡” 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑡𝑜
𝑙𝑖𝑠𝑡.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
- 𝐴 = {0, 2, 4, 6, 8, … , 100}
- 𝐵 = {𝑎, 𝑒, 𝐼, 𝑜, 𝑢}
- 𝐶 = {𝑥 ∶ 𝑥 is an integer, 1 < 𝑥 < 10}
𝑰𝒏𝒇𝒊𝒏𝒊𝒕𝒆 𝑺𝒆𝒕𝒔
𝐴𝑛 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆 𝒔𝒆𝒕 𝑖𝑠 𝑎 𝑠𝑒𝑡 𝑤𝑕𝑖𝑐𝑕 𝑖𝑠 𝑛𝑜𝑡 𝑓𝑖𝑛𝑖𝑡𝑒. 𝐼𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑡𝑜 𝑒𝑥𝑝𝑙𝑖𝑐𝑖𝑡𝑙𝑦 𝑙𝑖𝑠𝑡 𝑜𝑢𝑡 𝑎𝑙𝑙 𝑡𝑕𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑎𝑛 𝑖𝑛𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
- 𝑇 = {𝑥 ∶ 𝑥 𝑖𝑠 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 𝑥 > 100}
- 𝑄 = {-1,-2,-3,-4…}
- 𝑁 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠
- 𝐴 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠
𝑁𝑜𝑡𝑒: 𝑇𝑕𝑒 𝒏𝒖𝒎𝒃𝒆𝒓 𝒐𝒇 𝒆𝒍𝒆𝒎𝒆𝒏𝒕𝒔 𝑖𝑛 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡 𝐴 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝒏(𝑨).
𝑼𝒏𝒊𝒗𝒆𝒓𝒔𝒂𝒍 𝑺𝒆𝒕
𝐴 𝒖𝒏𝒊𝒗𝒆𝒓𝒔𝒂𝒍 𝒔𝒆𝒕 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑢𝑛𝑑𝑒𝑟 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟𝑎𝑡𝑖𝑜𝑛, 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝑐𝑎𝑝𝑖𝑡𝑎𝑙 U.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐺𝑖𝑣𝑒𝑛 𝑡𝑕𝑎𝑡 𝑈 = 5, 6, 7, 8, 9, 10, 11, 12 , 𝑙𝑖𝑠𝑡 𝑡𝑕𝑒
𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑠𝑒𝑡𝑠.
𝑎) 𝐴 = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑓𝑎𝑐𝑡𝑜𝑟 𝑜𝑓 60 = {5,6,10,12}
𝑏) 𝐵 = 𝑥 ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 = {5,7,11}

4. 𝑹𝒆𝒍𝒂𝒕𝒊𝒐𝒏𝒔 𝒐𝒇 𝒔𝒆𝒕𝒔
𝐸𝑞𝑢𝑎𝑙 𝑆𝑒𝑡𝑠
𝑇𝑤𝑜 𝑠𝑒𝑡𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 𝑖𝑓 𝑡𝑕𝑒𝑦 𝑐𝑜𝑛𝑡𝑎𝑖𝑛 𝑡𝑕𝑒 𝒔𝒂𝒎𝒆 𝒊𝒅𝒆𝒏𝒕𝒊𝒄𝒂𝒍 𝒆𝒍𝒆𝒎𝒆𝒏𝒕𝒔. 𝐼𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑕𝑎𝑣𝑒
𝑜𝑛𝑙𝑦 𝑡𝑕𝑒 𝑠𝑎𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠, 𝑡𝑕𝑒𝑛 𝑡𝑕𝑒 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑎𝑟𝑒 𝑂𝑛𝑒 − 𝑡𝑜 − 𝑂𝑛𝑒
𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒. 𝐸𝑞𝑢𝑎𝑙 𝑠𝑒𝑡𝑠 𝑎𝑟𝑒 𝑂𝑛𝑒 − 𝑡𝑜 − 𝑂𝑛𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 𝑏𝑢𝑡
𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 𝑠𝑒𝑡𝑠 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑎𝑙𝑤𝑎𝑦𝑠 𝑒𝑞𝑢𝑎𝑙 𝑠𝑒𝑡𝑠.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼
𝐶𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑡𝑕𝑒 𝑠𝑒𝑡𝑠: 𝑃 = {𝑇𝑜𝑚, 𝐷𝑖𝑐𝑘, 𝐻𝑎𝑟𝑟𝑦, 𝐽𝑜𝑕𝑛} , 𝑄 = {𝐷𝑖𝑐𝑘, 𝐻𝑎𝑟𝑟𝑦, 𝐽𝑜𝑕𝑛, 𝑇𝑜𝑚}
∴ 𝑃 𝑖𝑠 𝒆𝒒𝒖𝒂𝒍 𝑡𝑜 𝑄, 𝑎𝑛𝑑 𝑤𝑒 𝑤𝑟𝑖𝑡𝑒 𝑃 = 𝑄. 𝑇𝑕𝑒 𝑜𝑟𝑑𝑒𝑟 𝑖𝑛 𝑤𝑕𝑖𝑐𝑕 𝑡𝑕𝑒 𝑚𝑒𝑚𝑏𝑒𝑟𝑠 𝑎𝑝𝑝𝑒𝑎𝑟 𝑖𝑛 𝑡𝑕𝑒
𝑠𝑒𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑖𝑚𝑝𝑜𝑟𝑡𝑎𝑛𝑡.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼𝐼
𝑊𝑕𝑖𝑐𝑕 𝑜𝑓 𝑡𝑕𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑠𝑒𝑡𝑠 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙 𝑎𝑛𝑑 𝑤𝑕𝑖𝑐𝑕 𝑜𝑛𝑒𝑠 𝑎𝑟𝑒 𝑂𝑛𝑒 − 𝑡𝑜 − 𝑂𝑛𝑒
𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 ?
𝐴 = 𝑎, 𝑓, 𝑗, 𝑞 𝐵 = 1, 2, 3, 5, 8 𝐶 = 𝑥, 𝑦, 𝑧, 𝑤 𝐷 = {8, 1, 3, 5, 2}
𝑆𝑜𝑙𝑢𝑡𝑖𝑜𝑛
- 𝐵 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑒𝑞𝑢𝑎𝑙. 𝑇𝑕𝑒𝑦 𝑕𝑎𝑣𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠.
- 𝐴 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑂𝑛𝑒 − 𝑡𝑜 − 𝑂𝑛𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 𝑜𝑟 𝑚𝑎𝑡𝑐𝑕𝑖𝑛𝑔 𝑠𝑒𝑡𝑠. 𝐸𝑎𝑐𝑕 𝑠𝑒𝑡 𝑕𝑎𝑠
4 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠. 𝑇𝑕𝑒𝑦 𝑕𝑎𝑣𝑒 𝑡𝑕𝑒 𝑠𝑎𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑕𝑒 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠.
- 𝐵 𝑎𝑛𝑑 𝐷 𝑎𝑟𝑒 𝑂𝑛𝑒 − 𝑡𝑜 − 𝑂𝑛𝑒 𝑐𝑜𝑟𝑟𝑒𝑠𝑝𝑜𝑛𝑑𝑒𝑛𝑐𝑒 𝑎𝑛𝑑 𝑒𝑞𝑢𝑎𝑙 𝑠𝑒𝑡𝑠. 𝑇𝑕𝑒𝑦 𝑕𝑎𝑣𝑒 𝑡𝑕𝑒
𝑠𝑎𝑚𝑒 𝑖𝑑𝑒𝑛𝑡𝑖𝑐𝑎𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠.

5. 𝑺𝒖𝒃𝒔𝒆𝒕𝒔
𝐼𝑓 𝑒𝑣𝑒𝑟𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝐵 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎 𝑚𝑒𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝐴, 𝑡𝑕𝑒𝑛 𝑤𝑒 𝑠𝑎𝑦 𝐵 𝑖𝑠 𝑎 𝒔𝒖𝒃𝒔𝒆𝒕 𝑜𝑓 𝐴.
𝑊𝑒 𝑢𝑠𝑒 𝑡𝑕𝑒 𝑠𝑦𝑚𝑏𝑜𝑙 ⊂ or ⊆ 𝑡𝑜 𝑚𝑒𝑎𝑛 “𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓” .
𝑆𝑒𝑡 𝐵 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑠𝑒𝑡 𝐴 𝑖𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛: 𝐵 ⊂ 𝐴 𝑜𝑟 𝐵 ⊆ 𝐴
𝐴𝑛𝑑 𝑡𝑕𝑒 𝑠𝑦𝑚𝑏𝑜𝑙 ⊄ or ⊈ 𝑡𝑜 𝑚𝑒𝑎𝑛 “𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓”.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼
𝐴 = 1, 3, 5 , 𝐵 = 1, 2, 3, 4, 5
∴ 𝑆𝑜, 𝐴 ⊂ 𝐵 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑒𝑣𝑒𝑟𝑦 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑖𝑛 𝐴 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑖𝑛 𝐵.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼𝐼
𝑋 = 1, 3, 5 , 𝑌 = 2, 3, 4, 5, 6
∴ 𝑋 ⊄ 𝑌 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 1 𝑖𝑠 𝑖𝑛 𝑋 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑖𝑛 𝑌.
𝑁𝑜𝑡𝑒:
 𝐸𝑣𝑒𝑟𝑦 𝑠𝑒𝑡 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑖𝑡𝑠𝑒𝑙𝑓 𝑖. 𝑒. 𝑓𝑜𝑟 𝑎𝑛𝑦 𝑠𝑒𝑡 𝐴, 𝐴 ⊂ 𝐴
 𝑇𝑕𝑒 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑎𝑛𝑦 𝑠𝑒𝑡 𝐴 𝑖. 𝑒. Ø ⊂ 𝐴
 𝐹𝑜𝑟 𝑎𝑛𝑦 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵, 𝑖𝑓 𝐴 ⊂ 𝐵 𝑎𝑛𝑑 𝐵 ⊂ 𝐴 𝑡𝑕𝑒𝑛 𝐴 = 𝐵
𝑷𝒐𝒘𝒆𝒓 𝑺𝒆𝒕𝒔
𝑃𝑜𝑤𝑒𝑟 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑙𝑖𝑠𝑡 𝑎𝑙𝑙 𝑡𝑕𝑒 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑒𝑡.
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐿𝑖𝑠𝑡 𝑎𝑙𝑙 𝑡𝑕𝑒 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑄 = 𝑥, 𝑦, 𝑧
∴ 𝑇𝑕𝑒 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑜𝑓 𝑄 𝑎𝑟𝑒 { }, {𝑥}, {𝑦}, {𝑧}, {𝑥, 𝑦}, {𝑥, 𝑧}, {𝑦, 𝑧}𝑎𝑛𝑑 {𝑥, 𝑦, 𝑧}
𝑁𝑜𝑡𝑒:
𝑇𝑕𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑓𝑜𝑟 𝑎 𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡 𝐴 𝑖𝑠 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑡𝑕𝑒 𝑓𝑜𝑟𝑚𝑢𝑙𝑎:
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 = 2 𝑛 𝐴 𝑤𝑕𝑒𝑟𝑒 𝑛(𝐴) = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 𝑡𝑕𝑒 𝑓𝑖𝑛𝑖𝑡𝑒 𝑠𝑒𝑡 𝐴
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝑄 = 𝑥, 𝑦, 𝑧 . 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 𝑤𝑖𝑙𝑙 𝑄 𝑕𝑎𝑣𝑒?
∴ 𝑛 𝑄 = 3, 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑏𝑠𝑒𝑡𝑠 = 23 = 8 𝑜𝑟 𝑃(𝑄) = 23 = 8

6. 𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒔𝒆𝒕𝒔
𝑼𝒏𝒊𝒐𝒏 𝒐𝒇 𝑺𝒆𝒕𝒔
𝑇𝑕𝑒 𝒖𝒏𝒊𝒐𝒏 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠, 𝑤𝑕𝑖𝑐𝑕 𝑎𝑟𝑒 𝑖𝑛 𝐴 𝒐𝒓 𝑖𝑛 𝐵 𝒐𝒓 𝑖𝑛 𝑏𝑜𝑡𝑕.
𝐼𝑡 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴 ∪ 𝐵 𝑎𝑛𝑑 𝑖𝑠 𝑟𝑒𝑎𝑑 ‘𝐴 𝑢𝑛𝑖𝑜𝑛 𝐵’
𝐹𝑜𝑟𝑚𝑎𝑙𝑙𝑦: 𝐴 ∪ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵}
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐺𝑖𝑣𝑒𝑛 𝑈 = 1, 2, 3, 4, 5, 6, 7, 8, 10 , 𝑋 = 1, 2, 6, 7 𝑎𝑛𝑑 𝑌 = 1, 3, 4, 5, 8
𝐹𝑖𝑛𝑑 𝑋 ∪ 𝑌 ?
∴ 𝑋 ∪ 𝑌 = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 is written only once.
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔
𝑪𝒍𝒐𝒔𝒖𝒓𝒆:
− 𝑇𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑢𝑛𝑖𝑜𝑛.
− 𝑇𝑕𝑎𝑡 𝑖𝑠, 𝑡𝑕𝑒 𝑢𝑛𝑖𝑜𝑛 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑎 𝑠𝑒𝑡.
𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝑈𝑛𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑨 ∪ 𝑩 ∪ 𝑪 = 𝑨 ∩ 𝑩 ∪ 𝑪
𝑪𝒐𝒎𝒎𝒖𝒕𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝑈𝑛𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑨 ∪ 𝑩 = 𝑩 ∪ 𝑨
𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒚:
− 𝑇𝑕𝑒 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡, ∅, 𝑖𝑠 𝑏𝑜𝑡𝑕 𝑎 𝑟𝑖𝑔𝑕𝑡 𝑎𝑛𝑑 𝑙𝑒𝑓𝑡 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑓𝑜𝑟 𝑠𝑒𝑡 𝑢𝑛𝑖𝑜𝑛:
𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴, 𝑨 ∪ ∅ = 𝑨 𝑎𝑛𝑑 ∅ ∪ 𝑨 = 𝑨
𝐈𝐝𝐞𝐦𝐩𝐨𝐭𝐞𝐧𝐜𝐲:
− 𝑆𝑒𝑡𝑠 𝑎𝑟𝑒 𝑖𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑡 𝑢𝑛𝑑𝑒𝑟 𝑢𝑛𝑖𝑜𝑛: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴, 𝑨 ∪ 𝑨 = 𝑨
𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝑈𝑛𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑠 𝑜𝑣𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛: 𝑡𝑕𝑎𝑡 𝑖𝑠,
𝑨 ∪ (𝑩 ∩ 𝑪) = (𝑨 ∩ 𝑩) ∪ (𝑨 ∩ 𝑪)
𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝐴 ∪ 𝐵

7. 𝑰𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝑺𝒆𝒕𝒔
𝑇𝑕𝑒 𝒊𝒏𝒕𝒆𝒓𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑡𝑕𝑎𝑡 𝑎𝑟𝑒 𝑐𝑜𝑚𝑚𝑜𝑛 𝑡𝑜 𝑏𝑜𝑡𝑕
𝑠𝑒𝑡 𝐴 𝒂𝒏𝒅 𝑠𝑒𝑡 𝐵. 𝐼𝑡 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴 ∩ 𝐵 𝑎𝑛𝑑 𝑖𝑠 𝑟𝑒𝑎𝑑 ‘𝐴 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝐵’.
𝐹𝑜𝑟𝑚𝑎𝑙𝑙𝑦: 𝐴 ∩ 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∈ 𝐵}
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐺𝑖𝑣𝑒𝑛 𝑈 = 1, 2, 3, 4, 5, 6, 7, 8, 10 , 𝑋 = 1, 2, 6, 7 𝑎𝑛𝑑 𝑌 = 1, 3, 4, 5, 8
𝐹𝑖𝑛𝑑 𝑋 ∩ 𝑌 ?
∴ 𝑋 ∩ 𝑌 = {1}
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔
𝑪𝒍𝒐𝒔𝒖𝒓𝒆:
− 𝑇𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛.
− 𝑇𝑕𝑎𝑡 𝑖𝑠, 𝑡𝑕𝑒 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑎 𝑠𝑒𝑡.
𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑨 ∩ 𝑩 ∩ 𝑪 = 𝑨 ∩ 𝑩 ∩ 𝑪
𝑪𝒐𝒎𝒎𝒖𝒕𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑨 ∩ 𝑩 = 𝑩 ∩ 𝑨
𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒚:
− 𝑇𝑕𝑒 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑠𝑒𝑡, 𝑼, 𝑖𝑠 𝑏𝑜𝑡𝑕 𝑎 𝑟𝑖𝑔𝑕𝑡 𝑎𝑛𝑑 𝑙𝑒𝑓𝑡 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑓𝑜𝑟 𝑠𝑒𝑡 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛:
𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴, 𝑨 ∩ 𝑼 = 𝑨, 𝒂𝒏𝒅 𝑼 ∩ 𝑨 = 𝑨
𝑰𝒅𝒆𝒎𝒑𝒐𝒕𝒆𝒏𝒄𝒚:
− 𝑆𝑒𝑡𝑠 𝑎𝑟𝑒 𝑖𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑡 𝑢𝑛𝑑𝑒𝑟 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴, 𝑨 ∩ 𝑨 = 𝑨
𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑠 𝑜𝑣𝑒𝑟 𝑢𝑛𝑖𝑜𝑛: 𝑡𝑕𝑎𝑡 𝑖𝑠,
𝑨 ∩ (𝑩 ∪ 𝑪) = (𝑨 ∪ 𝑩) ∩ (𝑨 ∪ 𝑪)
𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝐴 ∩ 𝐵

8. 𝑫𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝑺𝒆𝒕𝒔
𝑇𝑕𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝐴 𝑤𝑕𝑖𝑐𝑕 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑎𝑙𝑠𝑜
𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝐵. 𝐼𝑡 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴 − 𝐵 .
𝐹𝑜𝑟𝑚𝑎𝑙𝑙𝑦: 𝐴 − 𝐵 = {𝑥|𝑥 ∈ 𝐴 𝑎𝑛𝑑 𝑥 ∉ 𝐵}
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐺𝑖𝑣𝑒𝑛 𝑈 = 1, 2, 3, 4, 5, 6, 7, 8, 10 , 𝑋 = 1, 2, 6, 7 𝑎𝑛𝑑 𝑌 = 1, 3, 4, 5, 8
𝐹𝑖𝑛𝑑 𝑋 − 𝑌 ?
∴ 𝑋 − 𝑌 = {2,6,7}
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔
𝑪𝒍𝒐𝒔𝒖𝒓𝒆:
− 𝑇𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒.
− 𝑇𝑕𝑎𝑡 𝑖𝑠, 𝑡𝑕𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑡𝑤𝑜 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑎 𝑠𝑒𝑡.
𝑨𝒔𝒔𝒐𝒄𝒊𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝒏𝒐𝒕 𝑎𝑠𝑠𝑜𝑐𝑖𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑖𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙,
𝐴– 𝐵 − 𝐶 ≠ 𝐴– 𝐵– 𝐶
𝑪𝒐𝒎𝒎𝒖𝒕𝒂𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝒏𝒐𝒕 𝑐𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑖𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙, 𝐴 − 𝐵 ≠ 𝐵 – 𝐴
𝑰𝒅𝒆𝒏𝒕𝒊𝒕𝒚:
− 𝑇𝑕𝑒 𝑒𝑚𝑝𝑡𝑦 𝑠𝑒𝑡, ∅, 𝑖𝑠 𝑎 𝑟𝑖𝑔𝑕𝑡 𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑓𝑜𝑟 𝑠𝑒𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒, 𝑏𝑢𝑡 𝑡𝑕𝑒𝑟𝑒 𝑖𝑠 𝑛𝑜 𝑙𝑒𝑓𝑡
𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦 𝑓𝑜𝑟 𝑠𝑒𝑡 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴, 𝐴 − ∅ = 𝐴
𝑰𝒅𝒆𝒎𝒑𝒐𝒕𝒆𝒏𝒄𝒚:
− 𝑆𝑒𝑡𝑠 𝑎𝑟𝑒 𝒏𝒐𝒕 𝑖𝑑𝑒𝑚𝑝𝑜𝑡𝑒𝑛𝑡 𝑢𝑛𝑑𝑒𝑟 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒: 𝑡𝑕𝑎𝑡 𝑖𝑠, 𝑖𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑠𝑒𝑡𝑠 𝐴,
𝑨 − 𝑨 = ∅ ≠ 𝑨
𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒗𝒊𝒕𝒚:
− 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑒𝑠 𝑜𝑣𝑒𝑟 𝑢𝑛𝑖𝑜𝑛: 𝑡𝑕𝑎𝑡 𝑖𝑠,
𝑨 ∩ (𝑩 ∪ 𝑪) = (𝑨 ∪ 𝑩) ∩ (𝑨 ∪ 𝑪)
𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝐴 − 𝐵

9. 𝑪𝒐𝒎𝒑𝒍𝒆𝒎𝒆𝒏𝒕 𝑶𝒇 𝑺𝒆𝒕𝒔
𝑇𝑕𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝐴 𝑖𝑠 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 𝑡𝑕𝑒 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑒 𝑤𝑕𝑖𝑐𝑕 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑖𝑛 𝐴.
𝐼𝑡 𝑖𝑠 𝑑𝑒𝑛𝑜𝑡𝑒𝑑 𝑏𝑦 𝐴′
𝐹𝑜𝑟𝑚𝑎𝑙𝑙𝑦: 𝐴′ = {𝑥|𝑥 ∉ 𝐴}
𝐸𝑥𝑎𝑚𝑝𝑙𝑒
𝐿𝑒𝑡 𝑈 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 , 𝑃 = {1, 2, 3}
𝑓𝑖𝑛𝑑 𝑃’ ?
∴ 𝑃′ = {4, 5, 6, 7, 8, 9, 10}
𝑷𝒓𝒐𝒑𝒆𝒓𝒕𝒊𝒆𝒔
𝑪𝒍𝒐𝒔𝒖𝒓𝒆:
−𝑇𝑕𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑠𝑒𝑡𝑠 𝑖𝑠 𝑐𝑙𝑜𝑠𝑒𝑑 𝑢𝑛𝑑𝑒𝑟 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛.
−𝑇𝑕𝑎𝑡 𝑖𝑠, 𝑡𝑕𝑒 𝑐𝑜𝑚𝑝𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎 𝑠𝑒𝑡 𝑖𝑠 𝑎 𝑠𝑒𝑡.
𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝐴′

10. 𝑽𝒆𝒏𝒏 − 𝑬𝒖𝒍𝒆𝒓′𝒔 𝒅𝒊𝒂𝒈𝒓𝒂𝒎
𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑖𝑠 𝑎 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑜𝑓 𝑜𝑣𝑒𝑟𝑙𝑎𝑝𝑝𝑖𝑛𝑔 𝑎𝑟𝑒𝑎𝑠 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑖𝑛𝑔 𝑠𝑒𝑡𝑠 𝑎𝑛𝑑 𝑡𝑕𝑒 𝑐𝑜𝑚𝑚𝑜𝑛
𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡𝑕𝑒𝑚.
𝑫𝒊𝒂𝒈𝒓𝒂𝒎𝒔
−𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚𝑠 𝑎𝑟𝑒 𝑛𝑜𝑡 𝑡𝑒𝑥𝑡𝑢𝑎𝑙.
𝑆𝑒𝑡 𝐴
U
𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑆𝑒𝑡
𝑨
− 𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚𝑠 𝑢𝑠𝑢𝑎𝑙𝑙𝑦 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑡𝑕𝑒 𝑢𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑠𝑒𝑡 𝑎𝑠 𝑎𝑠 𝑒𝑛𝑐𝑙𝑜𝑠𝑖𝑛𝑔
𝑟𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒, 𝑎𝑛𝑑 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑠𝑒𝑡𝑠 𝑎𝑠 𝑐𝑖𝑟𝑐𝑙𝑒𝑠 (𝑢𝑝 𝑡𝑜 𝑡𝑕𝑟𝑒𝑒).
𝑠𝑒𝑡 ∶ 𝐴 𝑠𝑒𝑡 ∶ 𝐴 ∩ 𝐵
𝑨 𝑩
𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑆𝑒𝑡 U 𝑠𝑒𝑡 ∶ 𝐵
Two Sets
𝑠𝑒𝑡 ∶ 𝐴 𝑠𝑒𝑡 ∶ 𝐴 ∩ 𝐶
𝑨 U
𝑠𝑒𝑡 ∶ 𝐴 ∩ 𝐵 𝑠𝑒𝑡 ∶ 𝐶
𝑩 𝑪
𝑠𝑒𝑡 ∶ 𝐵
𝑠𝑒𝑡 ∶ 𝐵 ∩ 𝐶
Three Sets
𝑠𝑒𝑡 ∶ 𝐴 ∩ 𝐵 ∩ 𝐶 𝑈𝑛𝑖𝑣𝑒𝑟𝑠𝑎𝑙 𝑆𝑒𝑡

11. 𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼 𝐴𝑛𝑠𝑤𝑒𝑟
15
𝑷
𝐿𝑒𝑡 𝑈 = {𝑥|15 ≤ 𝑥 ≤ 25, 𝑥 ∈ 𝑁} 17
𝑃 = {𝑠𝑒𝑡 𝑜𝑓 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 } 16 21 19
18 20
𝐷𝑟𝑎𝑤 𝑎𝑛𝑑 𝑙𝑎𝑏𝑒𝑙 𝑎 𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑡𝑜 23
22 24 U
𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑃 25
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼𝐼
𝐴𝑛𝑠𝑤𝑒𝑟
𝐷𝑟𝑎𝑤 𝑎𝑛𝑑 𝑙𝑎𝑏𝑒𝑙 𝑎 𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑡𝑜
𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡 𝑡𝑕𝑒 𝑠𝑒𝑡 𝑹
𝑅 = {𝑀𝑜𝑛𝑑𝑎𝑦, 𝑇𝑢𝑒𝑠𝑑𝑎𝑦, 𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦}.
𝑀𝑜𝑛𝑑𝑎𝑦
𝑇𝑢𝑒𝑠𝑑𝑎𝑦
𝑊𝑒𝑑𝑛𝑒𝑠𝑑𝑎𝑦
U
𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝐼𝐼𝐼
𝐺𝑖𝑣𝑒𝑛 𝑈 = 1, 2, 3, 4, 5, 6, 7, 8, 10 , 𝑋 = 1, 2, 6, 7 𝑎𝑛𝑑 𝑌 = 1, 3, 4, 5, 8
𝐹𝑖𝑛𝑑 𝑋 ∪ 𝑌 𝑎𝑛𝑑 𝑑𝑟𝑎𝑤 𝑎 𝑉𝑒𝑛𝑛 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 𝑡𝑜 𝑖𝑙𝑙𝑢𝑠𝑡𝑟𝑎𝑡𝑒 𝑋 ∪ 𝑌.
𝐴𝑛𝑠𝑤𝑒𝑟
𝑋 ∪ 𝑌 = {1, 2, 3, 4, 5, 6, 7, 8} ← 1 𝑖𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑜𝑛𝑙𝑦 𝑜𝑛𝑐𝑒.
𝑁𝑜𝑡𝑒: 𝐼𝑛 𝑔𝑒𝑛𝑒𝑟𝑎𝑙, 𝑡𝑕𝑒𝑟𝑒 𝑎𝑟𝑒 𝑚𝑎𝑛𝑦 𝑤𝑎𝑦𝑠 𝑡𝑕𝑎𝑡 3 𝑠𝑒𝑡𝑠 𝑚𝑎𝑦 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡.
𝑆𝑜𝑚𝑒 𝑒𝑥𝑎𝑚𝑝𝑙𝑒𝑠 𝑎𝑟𝑒 𝑠𝑕𝑜𝑤𝑛 𝑏𝑒𝑙𝑜𝑤.

12. 𝑾𝒐𝒓𝒌𝒊𝒏𝒈 𝒘𝒊𝒕𝒉 𝒕𝒘𝒐 𝒔𝒆𝒕𝒔
𝑬𝒍𝒆𝒎𝒆𝒏𝒕𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒖𝒏𝒊𝒐𝒏 𝒐𝒇 𝒕𝒘𝒐 𝒔𝒆𝒕𝒔
𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛 𝐵 − 𝑛(𝐴 ∩ 𝐵)
𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 − 𝐵 + 𝑛 𝐴 ∩ 𝐵 + 𝑛(𝐵 − 𝐴)
𝑛 𝐴 = 𝑛 𝐴− 𝐵 + 𝑛 𝐴∩ 𝐵 ⟶ 𝑛 𝐴 − 𝐵 = 𝑛 𝐴 − 𝑛(𝐴 ∩ 𝐵)
𝑛 𝐵 = 𝑛 𝐵− 𝐴 + 𝑛 𝐴∩ 𝐵 ⟶ 𝑛 𝐵 − 𝐴 = 𝑛 𝐵 − 𝑛(𝐴 ∩ 𝐵)
U
𝑨 𝑩
𝐷𝑖𝑠𝑗𝑜𝑖𝑛𝑡 𝑆𝑒𝑡𝑠
𝑛 𝐴 ∪ 𝐵 = 𝑛 𝐴 + 𝑛(𝐵)

13. 𝑾𝒐𝒓𝒌𝒊𝒏𝒈 𝒘𝒊𝒕𝒉 𝒕𝒘𝒐 𝒔𝒆𝒕𝒔
𝑬𝒍𝒆𝒎𝒆𝒏𝒕𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒖𝒏𝒊𝒐𝒏 𝒐𝒇 𝒕𝒉𝒓𝒆𝒆 𝒔𝒆𝒕𝒔
𝑛 𝐴∪ 𝐵∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴∩ 𝐵 − 𝑛 𝐴∩ 𝐶 − 𝑛 𝐵∩ 𝐶 + 𝑛 𝐴∩ 𝐵∩ 𝐶
𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 ∪ 𝐶 − 𝑛(𝐴 ∩ (𝐵 ∪ 𝐶))
𝑊𝑒 𝑕𝑎𝑣𝑒 ∶ 𝑛 𝐵 ∪ 𝐶 = 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐵 ∩ 𝐶
𝑃𝑢𝑡𝑡𝑖𝑛𝑔 𝑖𝑛 𝑡𝑕𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑓𝑜𝑟 "𝑛(𝐴 ∪ 𝐵 ∪ 𝐶)",
𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐵 ∩ 𝐶 − 𝑛(𝐴 ∩ (𝐵 ∪ 𝐶))
𝑊𝑒 𝑕𝑎𝑣𝑒 ∶ 𝑛 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝑛( 𝐴 ∩ 𝐵 ∪ 𝐴 ∩ 𝐶 )
𝑛 𝐴∩ 𝐵 ∪ 𝐴∩ 𝐶 = 𝑛 𝐴 ∪ 𝐵 + 𝑛 𝐴 ∪ 𝐶 − 𝑛((𝐴 ∩ 𝐵) ∩ (𝐴 ∩ 𝐶))
𝑛 𝐴∩ 𝐵 ∩ 𝐴∩ 𝐶 = 𝑛(𝐴 ∩ 𝐵 ∩ 𝐶)
𝑆𝑜, 𝑛 𝐴 ∩ 𝐵 ∪ 𝐶 = 𝑛 𝐴∪ 𝐵 + 𝑛 𝐴∪ 𝐶 − 𝑛 𝐴∩ 𝐵∩ 𝐶
𝑛 𝐴 ∪ 𝐵 ∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐵 ∩ 𝐶 − (𝑛 𝐴 ∪ 𝐵 + 𝑛 𝐴 ∪ 𝐶
− 𝑛 𝐴∩ 𝐵∩ 𝐶 )
𝑛 𝐴∪ 𝐵∪ 𝐶 = 𝑛 𝐴 + 𝑛 𝐵 + 𝑛 𝐶 − 𝑛 𝐴∩ 𝐵 − 𝑛 𝐴∩ 𝐶 − 𝑛 𝐵∩ 𝐶 + 𝑛 𝐴∩ 𝐵∩ 𝐶