AM10FM 2020 Tutorial 1 Tutorial 1 - Sets and Notation For the tutorial on 30th September 2020 1. Let A = {1, 2, aston}, B = {2, aston, 1, aston}, C = {2, 1} and D = {1, 2, pear}. (a) Is 1 ∈ A? (e) Is C ⊆ A? (b) Is 10 ∈ A? (f) Is C ⊂ A? (c) Is A = B? (g) Is A ⊆ C? (d) Is A = C? (h) Is D ⊆ B? 2. Given are the sets A = {m, a, t, h, e, m, a, t, i, c, s}, B = {a, s, t, o, n}, and C = {f, u, n, d, a, m, e, n, t, a, l}. Determine |A|, A ∩ B, A ∩ C, (A ∩ B) ∪ (A ∩ C), B ∪ C, A ∩ (B ∪ C) and {m, a, t, h, s}0 with universal set A. 3. Determine |A| when (a) A = {1, 2, 3}, (b) A = {{1, 2, 3}}, (c) A = {1, {2, 3}}, (d) A = {1, 2} × {2, 3}, (e) A = P({1, 2}). 4. Describe each of the following sets by listing their elements. Note: If the set has infinitely many elements, you may use “. . . ” to indicate the remaining elements of the set after you have listed sufficiently many elements to make the pattern they follow clear. (a) {x ∈ Z : −4 ≤ x < 5}, (b) {x ∈ Z : x2 < 50}, (c) {x ∈ N : x2 < 50}, (d) {x ∈ N : ∃m ∈ N such that x = 2m}, (e) { pq : p ∈ N and q = p + 1}, (f) {x : x is the name of a month containing “Q”}, (g) {x : x is a letter in “university”}, (h) {x : x is a day of the week with 10 letters}. 5. Put the correct sign (⇐, ⇒, or ⇔) between the following pairs of conditions/statements on a real number x ∈ R. Briefly explain your answers. 1 AM10FM 2020 Tutorial 1 (a) x ∈ N; 1/x ∈ Q. (c) 4x ∈ Z; (b) 3x ∈ Q; x ∈ Q. (d) x ∈ Q; x + 4 ∈ Z. 1/(1 − x) ∈ Q. 6. Find a concise way of describing each of the following sets by specifying a common property of their elements. (a) {. . . , −12, −8, −4, 0, 4, 8, 12, . . . } (b) {. . . , 81 , 14 , 12 , 1, 2, 4, . . . } (c) {. . . , −8, 4, −2, 1, − 21 , 14 , − 18 , . . . } (d) {3x : x ∈ Z} ∩ {2x : x ∈ Z} 7. (a) Out of the following sets, which has the most elements? (i) {x : x ∈ Z and 0 < x3 < 12345}, (ii) {x : x ∈ N and x < 100 and x is a multiple of 4}, (iii) {x3 : x ∈ Z and − 10 < x < 10}? (b) Find, if possible, infinite sets A and B such that A ∩ B = {0} and A ∪ B = Z. (c) Find, if possible, sets C and D such that C ∪ D = {b, i, g} and C ∩ D = {s, m, a, l, l}. 2