PROBLEM SETS 2:
1. 𝐿𝑒𝑡 𝐴 = {1,2} 𝑎𝑛𝑑 𝐵 = {3,4,5}. 𝑊𝑟𝑖𝑡𝑒 𝑑𝑜𝑤𝑛 𝑎𝑙𝑙 𝑡ℎ𝑒 𝑚𝑎𝑝𝑝𝑖𝑛𝑔𝑠 𝑓𝑟𝑜𝑚 𝐴 𝑡𝑜 𝐵.
2. 𝐻𝑜𝑤 𝑚𝑎𝑛𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑠 𝑎𝑟𝑒 𝑡ℎ𝑒𝑟𝑒 𝑓𝑟𝑜𝑚 𝑎 𝑠𝑒𝑡 𝑤𝑖𝑡ℎ 𝑚 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠 𝑖𝑛𝑡𝑜 𝑎 𝑠𝑒𝑡 𝑤𝑖𝑡ℎ 𝑛 𝑒𝑙𝑒𝑚𝑒𝑛𝑡𝑠?
3. 𝐿𝑒𝑡 𝑓 𝑏𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓: Ζ → Ζ: n → n + 1. Show that f is one − to − one and onto, and
down a formula for its reverse.
4. 𝐿𝑒𝑡 𝑁 𝑏𝑒 𝑡ℎ𝑒 𝑠𝑒𝑡 {0,1,2,3, … }𝑜𝑓 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠. 𝐿𝑒𝑡 𝑓: 𝑁 → 𝑁: 𝑛 → 𝑛 + 1. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓 𝑖𝑠 𝑜𝑛𝑒 −
𝑡𝑜 − 𝑜𝑛𝑒 𝑏𝑢𝑡 𝑖𝑠 𝑛𝑜𝑡 𝑜𝑛𝑡𝑜.
5. 𝐿𝑒𝑡 𝑓 𝑏𝑒 𝑡ℎ𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚 𝑁 𝑡𝑜 𝑍 𝑑𝑒𝑓𝑖𝑛𝑒 𝑏𝑦 𝑓(𝑛) = 𝑛 2 𝑖𝑓 𝑛 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑓(𝑛) = − 𝑛+1 2 𝑖𝑓 𝑛 𝑖𝑠
𝑜𝑑𝑑. 𝑆ℎ𝑜𝑤 𝑡ℎ𝑎𝑡 𝑓 𝑖𝑠 𝑎 𝑏𝑖𝑗𝑒𝑐𝑡𝑖𝑜𝑛, 𝑎𝑛𝑑 𝑓𝑖𝑛𝑑 𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑓𝑜𝑟 𝑖𝑡𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒.
6. 𝐿𝑒𝑡 𝐴 𝑏𝑒 𝑡ℎ𝑒 𝑜𝑝𝑒𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙 (0,1) = {𝑥 ∈ ℝ: 0 < 1𝑥 < 1}, 𝑎𝑛𝑑 𝑙𝑒𝑡 ℝ + 𝑏𝑒 𝑡ℎ𝑒 𝑠𝑒𝑡 𝑜𝑓 𝑎𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑟𝑒𝑎𝑙
𝑛𝑢𝑚𝑏𝑒𝑟𝑠. 𝑃𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑒 𝑓𝑢𝑐𝑛𝑡𝑖𝑜𝑛 𝑔𝑖𝑣𝑒𝑛 𝑏𝑦 𝑓: 𝐴 → ℝ+= 𝑥 → 1−𝑥 𝑥 𝑖𝑠 𝑎𝑛 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒, 𝑎𝑛𝑑 𝑤𝑟𝑖𝑡𝑒
𝑑𝑜𝑤𝑛 𝑎 𝑓𝑜𝑟𝑚𝑢𝑙𝑎 𝑓𝑜𝑟 𝑖𝑡𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒.
7. 𝐿𝑒𝑡 𝑓 𝑏𝑒 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦 𝑓: ℝ → ℝ: 𝑥 → 𝑥 3 .𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑤ℎ𝑒𝑡ℎ𝑒𝑟 𝑜𝑟 𝑛𝑜𝑡 𝑓 𝑖𝑠 𝑎𝑛 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑐𝑒, 𝑎𝑛𝑑 𝑖𝑓 𝑖𝑡
𝑖𝑠, 𝑓𝑖𝑛𝑑 𝑖𝑡𝑠 𝑖𝑛𝑣𝑒𝑟𝑠𝑒.