COS1501 Assignment 1 2021

COS1501 Assignment 1 2021



Question 1



ℤ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠.
ℤ = {… … … − 4, −3, −2, −1, 0, 1, 2, 3, … … … }



ℤ≥ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠.
ℤ≥ = {0, 1, 2, 3, … … … }
𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 ℤ≥ 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 ℤ
𝑆𝑜, ℤ≥ ⊆ ℤ



ℤ+ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑠𝑡𝑟𝑖𝑐𝑡𝑙𝑦 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠.
ℤ+ = {1, 2, 3, 4, … … … }
𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 ℤ+ 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 ℤ
𝑆𝑜, ℤ+ ⊆ ℤ
𝐴𝑙𝑠𝑜, 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛 ℤ+ 𝑎𝑟𝑒 𝑎𝑙𝑠𝑜 𝑒𝑙𝑒𝑚𝑒𝑛𝑡 𝑜𝑓 ℤ≥
𝑆𝑜, ℤ+ ⊆ ℤ≥



ℚ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
𝑎
𝑅𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑚𝑎𝑦 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏 ∈ ℤ , 𝑏 ≠ 0
𝑏


ℚ′ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
𝐼𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑟𝑒 𝑛𝑜𝑛 − 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑎𝑛𝑑 𝑛𝑜𝑛 − 𝑟𝑒𝑐𝑢𝑟𝑟𝑖𝑛𝑔 𝑑𝑒𝑐𝑖𝑚𝑎𝑙 𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛𝑠.
𝑎
𝐼𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑚𝑎𝑦 𝑁𝑂𝑇 𝑏𝑒 𝑒𝑥𝑝𝑟𝑒𝑠𝑠𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏 ∈ ℤ , 𝑏 ≠ 0
𝑏


ℝ 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
𝑇ℎ𝑖𝑠 𝑖𝑛𝑐𝑙𝑢𝑑𝑒𝑠 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑎𝑛𝑑 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
ℝ=ℚ∪ℚ′
𝑁𝑜𝑡𝑒 𝑡ℎ𝑎𝑡 ℚ ⊆ ℝ 𝑎𝑛𝑑 ℚ ′ ⊆ ℝ

, 1. ℤ≥ ⊆ ℤ 𝑇ℎ𝑖𝑠 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑇𝑅𝑈𝐸

2. ℤ+ ⊆ ℤ≥ 𝑇ℎ𝑖𝑠 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑇𝑅𝑈𝐸

3. ℝ ⊆ ℚ 𝑇ℎ𝑖𝑠 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝐹𝐴𝐿𝑆𝐸

𝑇ℎ𝑒 𝑎𝑏𝑜𝑣𝑒 𝑖𝑠 𝑠𝑡𝑎𝑡𝑖𝑛𝑔 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.
𝑇ℎ𝑖𝑠 𝑚𝑒𝑎𝑛𝑠 𝑡ℎ𝑎𝑡 𝑎𝑙𝑙 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑎𝑟𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙. (𝑊ℎ𝑖𝑐ℎ 𝑖𝑠 𝐹𝐴𝐿𝑆𝐸)
𝐶𝑜𝑢𝑛𝑡𝑒𝑟 𝑒𝑥𝑎𝑚𝑝𝑙𝑒. √12 𝑖𝑠 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟.
𝑁𝑜𝑤, √12 = 3.464101615137754587054892683011 … ……
√12 𝑖𝑠 𝑛𝑜𝑛 − 𝑡𝑒𝑟𝑚𝑖𝑛𝑎𝑡𝑖𝑛𝑔 𝑎𝑛𝑑 𝑛𝑜𝑛 − 𝑟𝑒𝑐𝑢𝑟𝑟𝑖𝑛𝑔. 𝑆𝑜, 𝑖𝑡 𝑖𝑠 𝑖𝑟𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙.
𝑊𝑒 ℎ𝑎𝑣𝑒 𝑎𝑛 𝑒𝑥𝑎𝑚𝑝𝑙𝑒 𝑜𝑓 𝑎 𝑟𝑒𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑖𝑛 ℚ . 𝑆𝑜, ℝ ⊈ ℚ


4. ℤ+ ⊆ ℝ 𝑇ℎ𝑖𝑠 𝑠𝑡𝑎𝑡𝑒𝑚𝑒𝑛𝑡 𝑖𝑠 𝑇𝑅𝑈𝐸
ℤ+ = {1, 2, 3, 4, … … … }
1 2 3 4
ℤ+ = { , , , , … … … }
1 1 1 1
𝑎
𝑆𝑜, 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 ℤ+ 𝑚𝑎𝑦 𝑏𝑒 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑜𝑟𝑚 𝑏
𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏 ∈ ℤ , 𝑏 ≠ 0

𝑆𝑜, 𝑎𝑙𝑙 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑜𝑓 ℤ+ 𝑎𝑟𝑒 𝑟𝑎𝑡𝑖𝑜𝑛𝑎𝑙.
𝑆𝑜, ℤ+ 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 ℚ 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑎 𝑠𝑢𝑏𝑠𝑒𝑡 𝑜𝑓 ℝ .



Question 1 THREE



Question 2



𝑇ℎ𝑒 𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑒𝑡𝑠 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑤𝑖𝑡ℎ 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑡ℎ𝑎𝑡 𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐴 𝑜𝑟
𝑏𝑒𝑙𝑜𝑛𝑔 𝑡𝑜 𝐵 𝑏𝑢𝑡 𝑛𝑜𝑡 𝑡𝑜 𝑏𝑜𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐵.
𝐴 + 𝐵 = {𝑥 | 𝑥 ∈ 𝐴 𝑜𝑟 𝑥 ∈ 𝐵 𝑏𝑢𝑡 𝑥 ∉ 𝐴 ∩ 𝐵}



𝐸𝑥𝑎𝑚𝑝𝑙𝑒 𝑖𝑓 𝐴 = {1, 2, 3, 4, } 𝑎𝑛𝑑 𝐵 = {3, 4, 5, 6, 7}
𝑇ℎ𝑒𝑛 𝐴 + 𝐵 = {1, 2, 5, 6, 7}



Question 2 FOUR