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