The roster form
1. Express the following sets using the roster form:
(a) {x ∈ N | 25 ≤ x2 < 50}
(b) {x ∈ Z | 25 ≤ x2 < 50} Hint: This set is different than (a).
(c) {3n : n ∈ Z and − 1 ≤ n ≤ 2}
(d) {3n ∈ Z : −1 ≤ n ≤ 2} Hint: This set is different than (c).
2. Use set builder notation to give a description of the set {−10, −5, 0, 5, 10, 15}.
3. List the members, i.e. elements, of the following sets.
(a) {x, {y}, {x, y}}
(b) P({x, {y}, {x, y}})
4. Let A = {1, 2, 5, 6, 7} and B = {2, 3, 4, 7, 8}. Find the following sets.
(a) A ∩ B
(b) A ∪ B
(c) A − B
(d) B − A
(e) A under the assumption that the universal set is U = {1, 2, 3, 4, 5, 6, 7, 8, 9}.
5. Use Venn diagrams to verify the identity A ⊕ B = (A ∪ B) − (A ∩ B). Moreover, clearly
identify each region of the identity in your responses. That is, don’t just identify the final regions
of the left and right hand sides; show how the final region is formed.
6. Let A = {1, 2, 3, 4} × {1, 2, 3}.
(a) What is the cardinality of A?
(b) Define the subset B of A as B = {(s, t) ∈ A : s ≤ t}. List the elements of B.
Sample Solution
(a) {25, 26, 27, 28, 29, 30, 31, 32}
(b) {-5,-3,-1 ,1 ,3 ,5}
(c) {0, 3, 6}
(d) {0 ,3 }
The roster form presents a set of elements in an organized list (Rosen 2019). In this case the sets given represent various subsets of numbers which can be expressed using the roster method. For example (a) is used to describe all natural numbers such that their square is greater than or equal to 25 and less than 50; in this case it would simply list each number meeting these criteria sequentially ({25, 26…32}) . Similarly set (b) expresses all integers that meet the same criteria although its outcome differs from before due to certain values being excluded from the scope of this particular subset therefore only those within range are listed({-5 … 5 }) . Set (c), represents all multiples of 3 within a certain range (-1 ≤ n ≤ 2 ) hence three entries appear on its roster ({ 0 , 3 , 6}) similarly for part d however since only even values are accepted its final result does contain fewer elements ({ 0 , 3} ).
In conclusion, the roster form allows us to quickly identify and visualize subsets containing numerical data by organizing them into easily readable lists; this eliminates any confusion regarding what specific terms apply when attempting to create such a representation.
