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Set Concepts
Summary: Sets help us to identify things such as objects, numerals, letters, symbols and so on in their respective groups depending on their characteristics or nature.
Introduction To Set Concepts
Sets help us to identify things such as objects, numerals, letters or symbols and so on in their respective groups depending on their characteristics or nature. Through understanding how various groups differ or how they are similar, we can easily sort out or relate such groups. This further leads to set applications that help us to solve problems in our daily life.
By the end of the study, the learners should be able to:
- Understand the concept and nature of sets, identify and organize to form into groups of items including numbers, letters, symbols, objects, ideas, etc. depending on their nature, characteristics, behavior, etc.
- Understand the different types and other categorizations of sets, their notations, comparisons and operations.
Sets
Meaning of a Set
A set is a collection of things with a common characteristic (s). A set can be made of various things such as objects, numerals, letters, words, symbols, plants, animals and so on. Members in a set are usually shown listed or defined with in a pair of curl brackets { }
If a set A = {a, b, c, d, e, f}, then the letters a, b, c, d, e, and f are members of the set A. Members of a set are also called elements.
Forming and naming of sets
All members of a given set must have a common characteristic(s) and they can be written as;
- a list of elements, e.g., V = {a, e, i, o, u},
- a phrase description, V = {all vowels of the English alphabets}, or
- a rule or trend given as a description all enclosed in a pair of curl brackets,
e.g. E = {all natural numbers divisible by 2}.
Example 1
List down the members of the set C, given;
C= { All Consonants of the English Alphabet }
Solution
C = { b, e, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z }
List down the members of the set W, if W = {all days of the week}
Solution
W = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
Given that, set A = {1, 2, 3, 4, 5, 6, …}, name set A.
Solution
A is a set of all counting (or natural) numbers.
Remember “…” , means “and so on”
P = {All even Prime numbers}
M = {Multiples of both 3 and 4 that are less than 50}
Qn2. Name the sets A = {1, 3, 5, … } and
B = {0, 2, 4, …100 }
Ans. 1: P = {2}
M = {12, 24, 36, 48}
Ans. 2: A = A set of odd numbers
B = A set of even numbers upto and including 100
Types of Sets
1. Finite and Infinite sets
A set with a countable number of elements is called a finite set. Such sets may include a set of mathematics books in a bookshelf, plates in a cupboard, Districts of Uganda and so on.
A set with an endless number of elements is called an infinite set. Such sets may include a set of all counting numbers, all odd numbers, all even numbers and so on.
Set A is a set of all trees in the world, state whether set A is finite or infinite.
Solution
Set A is finite, since all trees in the world can be counted, however much it may be difficult.
If a set P contains all odd numbers, state whether set P is finite or infinite.
Solution
Set P is finite, since odd numbers, form an endless list, that is 1, 3, 5, 7, …
A = {Factors of 200}
B = {Prime numbers less than 10}
C = {Multiples of 6}
D = {Letters of the English Alphabets}
Qn2. If set Q = {All numbers divisible by 99999}, state whether Q is finite or infinite.
Qn3. A set of numbers that are neither even nor odd is neither finite nor infinite. State true or false.
Ans.1: A = {1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 200}
B = {2, 3, 5, 7}
C = {6, 12, 18, 24, 30, 36 . . . }
D = {A, B, C, D, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}
Ans.2: Q is infinite
Ans.3: false
2. Empty sets
An empty set is a set which has no members. A set of cows with 7 legs each is an empty set, since no normal cow can have 7 legs. Empty sets are also called Null sets. The symbol for an empty set is { } or Ø. Such sets that are empty can be; snakes with legs, that is {snakes with legs} = Ø; Dogs that are goats, that is {Dogs that are goats} = Ø; Crying stones, that is {crying stones} = Ø and so on.
Set Notations
“Member of “ or “belongs to” (∈ )
A member of a set is one of the elements contained in that set. Given that set P = {a, b, c}, then letters a, b or c are each a member of set P, that is a ∈ P, b ∈ P and c ∈ P.
Letter d is not a member of set P and it expressed as d ∉ P.
“Number of”
“Number of” is a notation used to represent the number of elements in a given set. The number of members in set Q is represented as n(Q). If set Q = (u, v, w, x, y, z) then n(Q) = 6.
B = {m, o, f, i, e, l, d, s},
C = {1, 3, 6, 10, 15, 21} and
D = { }.
State true or false in each of the following:
a. a ∈ B b. n(A) = 6
c. n(B) = {8} d. n(D) = { }.
Ans.1: a. false b. true c. false d. false
“Contained in” or “subset of ”(⊂)
For two given sets one is said to be contained in the other if all the members in the first are all found in the second set.
A ⊂ B is read as “set A is contained in set B” or “set A is a sub set of set B”.
If set A = {0, 1, 2, 3, 4, 5} and B = {0, 1, 2, 3, …} then A ⊂ B, since all members of Set A are all found in set B.
“Contains “ or “Includes “or “is a superset of” ( ⊃ )
For two given sets one is said to contain the other if all the members in the second are also found in the first set. P ⊃ Q is read as “set P contains set Q” or “set P includes set Q” or “set P is a superset of set Q” If set P = {1, 3, 5, 7, 9} and set Q = {1, 5}, then P ⊃ Q since all members of set Q are also found in set P.
Remember that set Q is also called a subset of set P.
B = {4, 6, 8, 9, 10, 12, 14}, C = { } and
D = {1, 3, 6, 10, 15, 21}. State true or false in each of the following:
a. A ⊂ B b. B ⊂ C c. A ⊃ D
d. C ⊃ D e. {1, 3, 5} ⊂ D f. {7, 8, 9, 10} ⊂ A g. D ⊃ { }
Ans1: a. false b. false c. true d. true e. false f. true g. true
Given that set P = {1, 2, 3 … } and set Q = {4, 5, 6}, state true or false.
(i) P ⊂ Q (ii) Q ⊂ P (iii) P ⊃ Q (iv) Q ⊃ P
Solution
(i) P ⊂ Q is false (P is not a sub set of Q) (ii) Q ⊂ P is true ( Q is a subset of P)
(iii) P ⊃ Q is true (P is a superset of Q) (iv) Q ⊃ P is false (Q is not a superset of P)
A = {0, 1, 2, … } B = {1, 2, 3, ... } C = {1, 3, 5, … }
D = {0, 2, 4, … } E = {1, 4, 9, 16, 25} F = {1, 3, 6, 10, 15, 21}
State true of false for each of the following statements.
Qn1. a. E ⊂ B b. F ⊃ B c. E ⊃ D d. {1, 2, 3} ⊂ A
Qn2. a. {1, 3, 5} ⊂ B b. C ⊄ {1, 3, 6, 10} c. D ⊄ {2, 4, 6 . . . }
Ans1. a. true b. false c. false d. true
Ans2. a. true b. true c. true
Comparing sets
Equal (=) and Equivalent (≡) Sets
Equivalent sets are sets with the same number of members. If set R = {1, 2, 3, 4} and set S = {a, b. c. d}, then set R is equivalent to set S, that is, R ≡ S.
All equal sets are equivalent, for example, if P = {a, b, c} and Q = {c, b, a} then sets P and Q are both equal and equivalent. However not all equivalent sets are equal. For example, R = {1, 2, 3, 4} and set S = {a, b, c, d}. Sets R and S are equivalent, but not equal. Symbolically R ≡ S, but R ≠ S.
State whether the following sets are equal in each pair.
(i) A = {0, 2, 4, 6, 8} and B = {8, 6, 4, 2, 0} (iii) E = {1, 2, 3, 4, 5} and F = {1, 2, 4, 5, 6}
(ii) C = {a, b, c, e} and D = {b, c, e, f} (iv) G = {2, 3, 5, 7, 11} and H = {5, 3, 7, 2, 11}
Solution
(i) A = B, since all members in A are also in B
(ii) C ≠ D, since set C does not have element f and set D does not have element a.
(iii) E ≠ F, since set E does not have element 6 and set F does not have element 3.
(iv) G = H, since all the members in G are also in H.
a. A = { a, b, c, d} and B = {a, b, c, d} b. C = {u, v, w, y} and D = {u, v, w, x}
c. G = {1, 2, 3, 4, 5} and H = {5, 1, 2, 3, 4} d. I = {L, M, N, P, Q} and J = {1, 3, 5, 7, 9}
e. P = {8, 9, 11, 12} and Q = {A, B, C, D, E} f. R = {Red, Nice, Green} and S = {L, M, N}
Sub Sets
A set may have various sets formed by its members as used differently such sets which are formed are called subsets of the original set.
If we have a set of colours; C = {Red, Blue, Green}, we can have sub sets of set C as;
{ }, {Red}, {Blue}, {Green}, {Red, Blue}, {Red, Green}, {Blue, Green} and {Red, Blue, Green}.
| Listing of subsets of the set {a, b, c} | No. of elements in the set |
No. of formed subsets |
|---|---|---|
| The set { } has a subset { } | 0 |
1 |
| The set {a} has subsets { } and {a} | 1 |
2 |
| The set {a, b} has subsets { }, {a}, {b} and {a, b} | 2 | 4 |
| The set {a, b, c} has subset: { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} |
3 | 8 |
Using the subsets given study carefully the information in the following table. Complete the table below.
| Set | No. of elements, n | No. of subsets, N |
|---|---|---|
| { } | 0 | 1 |
| { a } | 1 | 2 |
| { a, b } | 2 | 4 |
| { a, b, c } | 3 | 8 |
| { a, b, c, d } | 4 | — |
| { a, b, c, d, e } | 5 | — |
| — | 6 | — |
From the table we realize that the number of subsets forms a pattern related to the number of elements in the set.
Complete the above pattern for N and check your answers in the table. Generally the number of elements, n, and the number of subsets, N, are related by the formula N = 2n.
Always remember that an empty set is a subset of all sets and every set can be a subset of itself, A subset, which does not contain all the members of the other set and is not empty, is called a proper subset.
For example, if set A = { 1, 2, 3} its subsets are: { }, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1,2,3}, with the sets: {1}, {2}, {3}, {1, 2} {1, 3},{2, 3} as the proper subsets of the set A.
So, a set with 3 elements has 6 proper subsets, Np, that is, Np = (2n) – 2
For, n = 3, Np = (2n) – 2
Np = (23) – 2
Np = 8 – 2
Np = 6.
Given that set Q = {W, X, Y, Z}:
(a) By Calculations, find;
(i) the number of subsets for set Q
(ii) the number of proper subsets for set Q
(b) Write down all the subsets of set Q
Solution
(a) (i) N = 2n, but n = 4
N = 24
N = 16 ∴ set Q has 16 subsets.
(ii) Np = (2n) – 2, but n = 4
Np = (24) – 2
Np = 16 – 2
Np = 14
∴ set Q has 14 proper subsets.
(b) Subsets of set Q are:
{ }, {W}, {X}, {Y}, {Z}, {W, X}, {W, Y}, {W, Z}, {X, Y}, {X, Z}, {Y, Z}, {W, X, Y}, {W, X, Z}, {W, Y, Z}, {X, Y, Z} and {W, X, Y, Z}.
Qn2. Write down all the proper sub sets of: a. C = {r, s, t} b. {pencil, pen, book}
Qn3. List all the subsets of the set E = {2, 4, 6, 8}.
Universal sets
Given a group, various subgroups can be obtained from it or as part of it. For a major group of scholastic materials subgroups of pens, pencils, rulers, papers, books, file folders and so on may be included. Such a major group is called the Universal set. A Universal set is denoted: ξ.
Study the other sets that can be included in the universal set; Furniture
| Universal Set | Included Sets |
|---|---|
| Furniture | {Chairs, Tables, Desks, Benches}, {Cupboards, Shelves, Racks, Wardrobes}, {Doors, Windows, Vents}, etc. |
| Scholastic Materials | {Pens, Pencils, Crayons, Markers}, Materials {Graph books, Exercise books, Papers}, {Rulers, Compasses, Dividers, Set squares, Protractors} |
A Universal set; East Africa includes elements: Uganda, Kenya and Tanzania.
Operations on sets
Union of sets (∪)
Union of sets refers to the combination of 2 (or more) sets to form a single set that contains all the members of the original sets. Union of sets is denoted, ∪; where, for set P union Q,
we write P ∪ Q. Always remember that a member found in more than one of the original sets need only be shown once in the union.
Given that, P = {2, 3, 5, 7, 11}, S = {1, 4, 9, 16} and T = {1, 3, 6, 10, 15}. Find P U S U T
Solution
P U S U T = {1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 15, 16}
Given that X U Y = {2, 3, 4, 5, 6} and X = {2, 4, 6}. Find set Y if 2 is the only member common for both sets X and Y.
Solution
Y = {2, 3, 5, 7}
Given that sets; A = {1, 2, 3, 4, 5, 6, 7, 8} B = {1, 3, 5, 7, 9} C = { a, b, c, d, e }
D = {a, e, i, o, u} E = {a, b, c, d, o, p, r, u} F = {1, 3, 6, 10, 15, 21}
Qn1. Find: a. A ∪ B b. C U D
Qn2. Find: a. A ∪ B ∪ F b. C ∪ D ∪ E
Intersection of sets (∩)
Intersection of 2 (or more) sets is a single set made containing only members which are common to both (or all) the original sets. For set P intersection Q, we write P ∩ Q.
Given that, set V = {a, e, i, o, u} and set B = {a, b, c, d, k, l, o, r}. Find V ∩ B
Solution
V ∩ B = {a, o}
Example 2
Given that, set N = {1, 2, 3, 4, 5, 6} and set P = {2, 3, 5, 7}. Find N ∩ P.
Solution
N ∩ P = { 2, 3, 5}
Qn. Given the sets;
A = {1, 2, 3, 4, 5, 6, 7, 8} F = {Red, Blue, Green}
B = {2, 3, 5, 7, 11} H = {1, 3, 5, 7, 9, 11, 13}
E = {Black, Yellow, Red}
J = {Red, Orange, Yellow, Green, Blue, Indigo, Violet}
Find: a. A ∩ B ∩ H b. E ∩ F ∩ J
Disjoint sets
If 2 (or more) sets have no members in common they are said to be disjoint. P and Q are disjoint sets if their intersection is an empty set, that is, P and Q = { }.
If P = {all even numbers} and Q = {all odd numbers}. We can have P = {0, 2, 4, 6, … } and Q = {1, 3, 5, 7 … }, giving P ∩ Q = { }, meaning that P and Q are disjoint sets.
A Complement set
A complement set is a set with members that are not found in a given set, but are in its universal set. A complement of set A is denoted A´, if a universal set ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9} and set A = {2, 3, 5, 7}, then the complement of set A is, A´ = {1, 4, 6, 8, 9}
Difference of sets
The difference of 2 sets is the set of elements belonging to the first set but not to the second set. P difference Q is denoted P – Q.
If set A = {a, b, c, d, e} and set B = {a, e, i, o, u }, then A – B = {b, c, d}
Qn1. State whether each of the following pair of sets is disjoint or not;
a. A = {a, b, c, d} and B = {e, f, g, h, i} b. C = {1, 2, 3, 4, 5} and D = {1, 3, 6, 10}
Qn2. Given that ξ = {first ten prime numbers} and P = {13, 17, 19, 23}, find P5
Qn3. Given that set P = {1, 2, 3, 4, 5} and set Q = {a, b, d, o, r}, find P – Q.
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This work is licensed by Mr. Luutu Suleiman under a Creative Commons Attribution License (CC-BY 2.0), and is an Open Educational Resource.
Last edited by Mr. Luutu Suleiman on May 1, 2010 15:14