MAT111 Tutorial 1 PDF

Level 1 Semester 1 Faculty of Science MAT111 Tutorial 1 1. Classify each if the following numbers (ℕ, ℤ, ℚ, ℚ𝑐 , ℝ): i. 2 ii. 1.333333 … iii. 2 iv. −0.5 v. − 16 1 vi. 2 5 vii. −7...

Level 1 Semester 1 Faculty of Science MAT111 Tutorial 1 1. Classify each if the following numbers (ℕ, ℤ, ℚ, ℚ𝑐 , ℝ): i. 2 ii. 1.333333 … iii. 2 iv. −0.5 v. − 16 1 vi. 2 5 vii. −7 Natural number ℕ = *1,2,3,4, … + Integers ℤ = *… , −2, −1,0,1,2, … + 𝑎 Rational numbers ℚ = *𝑏 : 𝑎, 𝑏 ∈ ℤ, 𝑏 ≠ 0+ Irrational numbers ℚ𝑐 = ℝ − ℚ Real numbers ℝ = *𝑥: −∞ < 𝑥 < ∞+ 2. List the elements of each of the following sets, using the ‘...’ notation where necessary: i. *𝑥: 𝑥 is integer and − 3 < 𝑥 < 4+ ii. 𝑥: 𝑥 is a positive integer multiple of three iii. *𝑥: 𝑥 is an integer smaller than 5+ iv. *𝑥: 𝑥 is a rational number with denomenator 2+  The set *𝑥: 𝑥 = 𝑦 2 and 𝑦 is an integer+ can be represented in the form … a) *1,2,3,4, … + b) *1,4,9,16, … + c) *0,1,4,9,16, … + d) *… , −9, −4, −1,0,1,4,9, … +  The set *𝑥: 𝑥 is an integer and (3𝑥 − 1)(𝑥 + 2) = 0+ can be represented in the form … 1 a) * , −2+ 𝟏 b) , −𝟐 c) *−𝟐+ d) −𝟐 3 𝟑  The set*𝑥: 2𝑥 is a positive integer+ can be represented in the form … a) *𝟐, 𝟒, 𝟔, 𝟖, … + b) *𝟏, 𝟐, 𝟑, 𝟒, … + 𝟏 𝟑 𝟓 c) * , 𝟏, , 𝟐, , 𝟑, … + 𝟏 𝟑 d) *𝟎, ±. ±𝟏, ± , … + 𝟐 𝟐 𝟐 𝟐 𝟐  If 𝑋 = 0,1,2 , then the set *𝑧: 𝑧 ∈ 𝑋 or − 𝑧 ∈ 𝑋+ equals … a) *0,1,2+ b) *0+ c) *0, −1, −2+ d) *0,1, −1,2, −2+  The set*𝑥: 𝑥 is an integer and 1/8 < 𝑥 < 17/2+ has the cardinality … a) ∞ b) 𝟎 c) 𝟖 d) 𝟑  The set {3,6,9,12,15,...,27,30+ can be represented in the form … a) *𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟, 3 ≤ 𝑥 ≤ 30+ b) *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3* c) *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑 𝑜𝑟 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3+ d) *𝑥: 𝑥 = 3𝑧 𝑎𝑛𝑑 1 ≤ 𝑧 ≤ 10+  The set {2,3,5,7,11,13,17,19,23,...+ can be represented as … a) *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟+ b) *𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑟𝑖𝑚𝑒 𝑛𝑢𝑚𝑏𝑒𝑟+ b) *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑 𝑜𝑟 𝑝𝑟𝑖𝑚𝑒+ d) *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑 𝑎𝑛𝑑 𝑝𝑟𝑖𝑚𝑒+  State whether each of the following statements is true or false. i. 2 ∈ *1,2,3,4,5+ ii. *2+ ∈ *1,2,3,4,5+ iii. 2 ⊆ *1,2,3,4,5+ iv. *2+ ⊆ *1,2,3,4,5+ v. ∅ ⊆ *∅ , *∅ ++ vi. *∅+ ⊆ *∅, *∅++ vii. 0 ∈ ∅ viii. *1,2,3,4,5+ = *5,4,3,2,1+  List all subsets of the sets 𝑎 , 𝑎, 𝑏 , *𝑎, 𝑏, 𝑐+. Can you predict how many subsets a set with 𝑛 elements will have? Does the empty set have any subsets?  In each of the following cases state whether 𝑥 ∈ 𝐴, 𝑥 ⊆ 𝐴, both or neither: (i) 𝑥 = *1+; 𝐴 = *1,2,3+ (ii) 𝑥 = *1+; 𝐴 = **1+, *2+, *3++ (iii) 𝑥 = *1+; 𝐴 = *1,2, *1,2++ (iv) 𝑥 = *1,2+; 𝐴 = *1,2, *1,2++ (v) 𝑥 = *1+; 𝐴 = **1,2,3++ (vi) 𝑥 = 1; 𝐴 = **1+, *2+, *3++  Let ℧ = 𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 2 ≤ 𝑥 ≤ 10 , 𝐴 = *𝑥: 𝑥 𝑖𝑠 𝑜𝑑𝑑+ and 𝐵 = *𝑥: 𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3+, then a) 𝐴 ⊂ 𝐵 b) 𝐵 ⊂ 𝐴 c) 𝐴 = 𝐵 d) niether  Let ℧ = 𝑥: 𝑥 𝑖𝑠 𝑎𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑎𝑛𝑑 2 ≤ 𝑥 ≤ 10 , 𝐴 = 𝑥: 𝑥 ∈ ℤ and 𝐵 = *𝑥: 𝑥 𝑖𝑠 𝑎 𝑝𝑜𝑤𝑒𝑟 𝑜𝑓 2 𝑜𝑟 3+. Then a) 𝐴 ⊂ 𝐵 b) 𝐵 ⊂ 𝐴 c) 𝐴 = 𝐵 d) niether  Among 18 students in a room, 7 study mathematics, 10 study science, and 10 study computer programming. Also, 3 study mathematics and science, 4 study mathemati cs and computer programming, and 5 study science and computer programming. W e know that 1 student studies all three subjects. How many of these students study none of the three subjects? Operations on sets 4. If ℧ = *1,2,3,4,5,6,7,8,9,10+ 𝐴 = 𝑥 ∈ ℧: 𝑥 is less than 7 , 𝐵 = *𝑥 ∈ ℧: 𝑥 𝑖𝑠 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 3+ then compute the following: i. 𝐴 ∪ 𝐵 ii. 𝐴 ∩ 𝐵 iii. 𝐴𝑐 or 𝐴 iv. 𝐵𝑐 or 𝐵 v. 𝐴\B 5. Draw Venn diagrams and shade the regions representing each of the following sets: i. 𝐴 ∪ 𝐵 ii. 𝐴 ∩ 𝐵 iii. 𝐴𝑐 iv. 𝐴 ∩ 𝐵 𝑐 v. 𝐴 ∪ 𝐵 vi. 𝐴 ∪ 𝐵 ∪ 𝐶 vii. 𝐴 ∩ (𝐵 ∩ 𝐶)