MAT2611 SEMESTER 1 ASSIGNMENT 1 2022
Problem 1
Addendum A
Exercise 2.8
𝐴 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟} 𝑎𝑛𝑑 𝐵 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟}
1.
ℕ = {1, 2, 3, 4, 5, 6, 7, … }
𝐴 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟} 𝑎𝑛𝑑 𝐵 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟}
𝐴 = {2, 3, 5, 7, 11, 13,17, 19, … } 𝑎𝑛𝑑 𝐵 = {1, 3, 5, 7, 9, 11, 13, 15, … }
2 ∈ 𝐵 𝑏𝑢𝑡 2 ∉ 𝐵
𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐴 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐵
∴ 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵
9 ∈ 𝐵 𝑏𝑢𝑡 9 ∉ 𝐴
𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐵 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐴
∴ 𝐵 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴
2.
𝐴≠𝐵
𝑇ℎ𝑒 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 𝑒𝑞𝑢𝑎𝑙 𝑖𝑓 𝐴 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵 𝐴𝑁𝐷 𝐵 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴
𝐹𝑟𝑜𝑚 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 2.8 𝑝𝑎𝑟𝑡 1, 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵 𝐴𝑁𝐷 𝐵 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴.
𝐴𝑙𝑠𝑜, 𝑒𝑞𝑢𝑎𝑙 𝑠𝑒𝑡𝑠 ℎ𝑎𝑣𝑒 𝑒𝑥𝑎𝑐𝑙𝑡𝑦 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠.
15 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐵 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐴. 𝑇ℎ𝑒 𝑠𝑒𝑡𝑠 𝑑𝑜 𝑛𝑜𝑡 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
𝑆𝑜, 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐵.
Problem 1
Addendum A
Exercise 2.8
𝐴 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟} 𝑎𝑛𝑑 𝐵 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟}
1.
ℕ = {1, 2, 3, 4, 5, 6, 7, … }
𝐴 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟} 𝑎𝑛𝑑 𝐵 = {𝑥 ∈ ℕ ∶ 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑛𝑢𝑚𝑏𝑒𝑟}
𝐴 = {2, 3, 5, 7, 11, 13,17, 19, … } 𝑎𝑛𝑑 𝐵 = {1, 3, 5, 7, 9, 11, 13, 15, … }
2 ∈ 𝐵 𝑏𝑢𝑡 2 ∉ 𝐵
𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐴 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐵
∴ 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵
9 ∈ 𝐵 𝑏𝑢𝑡 9 ∉ 𝐴
𝑇ℎ𝑒𝑟𝑒 𝑖𝑠 𝑎𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐵 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐴
∴ 𝐵 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴
2.
𝐴≠𝐵
𝑇ℎ𝑒 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑤𝑜𝑢𝑙𝑑 𝑏𝑒 𝑒𝑞𝑢𝑎𝑙 𝑖𝑓 𝐴 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵 𝐴𝑁𝐷 𝐵 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴
𝐹𝑟𝑜𝑚 𝐸𝑥𝑒𝑟𝑐𝑖𝑠𝑒 2.8 𝑝𝑎𝑟𝑡 1, 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐵 𝐴𝑁𝐷 𝐵 𝑖𝑠 𝑛𝑜𝑡 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝐴.
𝐴𝑙𝑠𝑜, 𝑒𝑞𝑢𝑎𝑙 𝑠𝑒𝑡𝑠 ℎ𝑎𝑣𝑒 𝑒𝑥𝑎𝑐𝑙𝑡𝑦 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠.
15 𝑏𝑒𝑙𝑜𝑛𝑔𝑠 𝑡𝑜 𝐵 𝑏𝑢𝑡 𝑑𝑜𝑒𝑠 𝑛𝑜𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐴. 𝑇ℎ𝑒 𝑠𝑒𝑡𝑠 𝑑𝑜 𝑛𝑜𝑡 ℎ𝑎𝑣𝑒 𝑡ℎ𝑒 𝑠𝑎𝑚𝑒 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠
𝑆𝑜, 𝐴 𝑖𝑠 𝑛𝑜𝑡 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝐵.
