Sets
Digital Handwritten Lesson
New Millennium Academy
Birauta, Pokhara-17, Kaski, Nepal
Definition of a Set
A set is a well-defined collection of distinct objects. These objects are called the elements or members of the set.
Sets are denoted by capital letters: A, B, C, …, X, Y, Z.
Elements are written inside curly braces — they may be letters, numbers, names, or any well-defined objects.
- \(A = \{a, e, i, o, u\}\)→A set of vowels
- \(B = \{2, 4, 6, 8, 10\}\)→First five even numbers
- Well-defined: There must be a clear rule to decide whether any object belongs to the set. "The collection of tall people" is not a set (subjective). "People over 6 feet tall" is a set.
- Distinct: Each object is unique. If an element is listed more than once, it is still one member.
- Order-independent: \(\{1, 2, 3\}\) is the same set as \(\{3, 1, 2\}\).
Types of Sets
A. Empty Set (Null Set)
- \(A = \{\text{seven feet tall boy in your class}\}\)
- \(B = \{\text{8th colour in a rainbow}\}\)
Sets \(A\) and \(B\) contain no elements at all.
The set with no elements is called the empty set (or null set).
It is denoted by ∅ (phi) or { }.
Let \(A = \{0\}\). Set \(A\) is not an empty set — it contains the element "0".
B. Finite and Infinite Sets
- List 1: \(A = \{\text{vowels of the English alphabet}\}\)
- List 2: \(B = \{\text{even numbers}\}\)
- \(A = \{a, e, i, o, u\}\) — we can count elements of set \(A\).
- \(B = \{2, 4, 6, 8, 10, \ldots\}\) — we cannot count elements of \(B\); there is no end.
A set having a fixed (countable) number of elements.
Example: \(A = \{a, e, i, o, u\}\)
A set that does not have a fixed number of elements (goes on without end).
Example: \(B = \{2, 4, 6, 8, \ldots\}\)
C. Equal and Equivalent Sets
\(X = \{1, 2, 3, 4\}\)
\(Y = \{3, 1, 2, 4\}\)
\(P = \{\text{apple, banana, mango}\}\)
\(Q = \{\text{red, blue, green}\}\)
- Group 1 (X and Y): same elements → Equal Sets
- Group 2 (P and Q): different elements but same number of elements → Equivalent Sets
Two non-empty sets \(A\) and \(B\) are equal if they contain exactly the same elements.
Two non-empty sets \(P\) and \(Q\) are equivalent if they have the same number of elements.
D. Universal Set
- \(S_1 = \{\text{girls of class 7}\}\)
- \(S_2 = \{\text{boys of class 7}\}\)
- \(S_3 = \{\text{students of class 7}\}\)
\(S_3\) contains all elements of both \(S_1\) and \(S_2\).
\(S_3\) is the universal set of \(S_1\) and \(S_2\).
- \(S_1 = \{2, 4, 6, 8\}\), \(S_2 = \{1, 3, 6, 9\}\), \(S_3 = \{1, 5, 7, 10\}\)
- \(S_4 = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\)
\(S_4\) is the universal set of \(S_1\), \(S_2\), and \(S_3\).
A universal set is the master set that contains all elements of all sets under consideration. It is generally written as \(U\).
Subsets, Proper Subsets, and Improper Subsets
Let \(A = \{1, 2, 3\}\). All sets that can be formed using only elements of \(A\) are listed below:
Gold border = improper subset | Grey = empty set (like air — present everywhere)
- \(S_1\)–\(S_8\) are all made from elements of \(A\) only.
- \(S_1\)–\(S_7\) are not equal to \(A\). → Proper Subsets
- \(S_8\) is equal to \(A\). → Improper Subset
A set within a set is called a subset.
Set \(B\) is a proper subset of \(A\) if:
- \(B\) contains elements of \(A\) only, and
- \(B \neq A\)
Set \(B\) is an improper subset of \(A\) if:
- \(A = B\)
Every set is an improper subset of itself.
- The empty set \(\emptyset\) is a subset of every set.
- Each set has exactly one improper subset (itself); all other subsets are proper subsets.
Formula for Number of Subsets
| Set | No. of elements (\(n\)) | All Subsets | No. of subsets |
|---|---|---|---|
| \(\{a\}\) | 1 | \(\{a\},\ \{\}\) | \(2 = 2^1\) |
| \(\{a, b\}\) | 2 | \(\{a\},\ \{b\},\ \{a,b\},\ \{\}\) | \(4 = 2^2\) |
| \(\{a, b, c\}\) | 3 | \(\{a\},\ \{b\},\ \{c\},\ \{a,b\},\ \{b,c\},\ \{a,c\},\ \{\},\ \{a,b,c\}\) | \(8 = 2^3\) |
| — | — | — | — |
| \(\{a, b, c, \ldots\}\) | \(n\) | — | \(2^n\) |
\(\text{Proper subsets} + \text{Improper subsets} = 2^n\)
\(\text{Proper subsets} + 1 = 2^n\)
\(\therefore\ \text{No. of proper subsets} = 2^n - 1\)
Course material curated by Mr. Nripendraswar Acharya