01 · Explore
Introduction to Sets and Representations
The concept of a set is a fundamental building block of modern mathematics, used to define relations, functions, and probability. Developed by Georg Cantor, a set is a well-defined collection of objects.
A collection is 'well-defined' if we can definitely decide whether a particular object belongs to it or not. For example, 'the vowels in the English alphabet' is a set because we know exactly which letters (a, e, i, o, u) are included. However, 'the five most renowned mathematicians' is not a set because the criteria for 'renowned' varies from person to person.
In set notation, we use capital letters (A, B, C) to denote sets and small letters (a, b, c) for their elements. The symbol ∈ (epsilon) means 'belongs to'. If 'a' is an element of set A, we write a ∈ A. If it is not, we write a ∉ A.
| Symbol | Meaning |
|---|---|
| N | The set of all natural numbers |
| Z | The set of all integers |
| Q | The set of all rational numbers |
| R | The set of all real numbers |
| Z+ | The set of all positive integers |
| Q+ | The set of all positive rational numbers |
| R+ | The set of all positive real numbers |
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Question
Is the collection of 'all tall students in your class' a set?
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02 · Explore
Roster and Set-Builder Forms
There are two primary ways to represent a set: the Roster (Tabular) form and the Set-Builder form.
In Roster form, all elements are listed within braces { } and separated by commas. The order of elements does not matter, and elements are typically not repeated. For example, the set of letters in the word 'SCHOOL' is {S, C, H, O, L}.
In Set-Builder form, we describe the elements by a common property. We use a variable (like x) followed by a colon ':' (which means 'such that') and then state the property. For example, V = {x : x is a vowel in English alphabet}.
Converting to Roster Form
A = {x : x is a positive integer and x² < 40}
We check positive integers: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36. 7²=49, which is > 40. So the elements are 1, 2, 3, 4, 5, 6. Roster form: {1, 2, 3, 4, 5, 6}.
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Question
Write the set of all even natural numbers less than 10 in roster form.
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03 · Explore
Empty, Finite, and Infinite Sets
Sets are categorized based on the number of elements they contain.
An Empty Set (or null/void set) contains no elements at all. It is denoted by the symbol φ or empty braces { }. For example, the set of natural numbers between 1 and 2 is an empty set.
A Finite Set contains a definite number of elements. The number of distinct elements in a finite set S is denoted by n(S). If a set has an unending number of elements, it is called an Infinite Set. The set of all natural numbers {1, 2, 3, ...} is infinite.
Two sets A and B are 'Equal' if they have exactly the same elements, regardless of the order or repetition. If A = {1, 2} and B = {2, 2, 1}, then A = B.
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Question
Is the set of all points on a line segment finite or infinite?
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04 · Explore
Subsets and Intervals of R
A set A is a subset of B if every element of A is also in B.
We write A ⊂ B to denote that A is a subset of B. If A ⊂ B and A ≠ B, then A is a 'proper subset' and B is the 'superset'. Every set is a subset of itself, and the empty set φ is a subset of every set.
For real numbers, we use intervals. An 'Open Interval' (a, b) includes all numbers between a and b but not a or b themselves. A 'Closed Interval' [a, b] includes the endpoints. We can also have semi-open intervals like [a, b) or (a, b].
The Universal Set (U) is the basic set that contains all objects under consideration in a particular context. For example, when studying integers, the set of real numbers R can be the universal set.
Interval Notation
{x : x ∈ R, -4 < x ≤ 6}
The value -4 is excluded (indicated by <), and 6 is included (indicated by ≤). In interval form, this is written as (-4, 6].
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If A = {1, 2} and B = {1, 2, 3}, is A ⊂ B?
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05 · Explore
Venn Diagrams and Union of Sets
Venn diagrams represent set relationships visually using rectangles for the universal set and circles for subsets.
The Union of two sets A and B (A ∪ B) is the set of all elements that are in A, or in B, or in both. Common elements are listed only once. For example, if A = {2, 4, 6} and B = {6, 8, 10}, then A ∪ B = {2, 4, 6, 8, 10}.
The operation of union follows the Commutative law (A ∪ B = B ∪ A) and the Associative law ((A ∪ B) ∪ C = A ∪ (B ∪ C)). The empty set φ acts as the identity element for union (A ∪ φ = A).
The Union Operation
- 1
Set A and Set B
Identify all elements present in both sets.
- 2
Combine Elements
List every element from A and every element from B.
- 3
Remove Duplicates
Ensure elements common to both are written only once.
- 4
Result: A ∪ B
The final collection represents the union.
A visual flow of how the union of two sets is formed by combining all unique members.
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If A ⊂ B, what is A ∪ B?
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06 · Explore
Intersection and Difference of Sets
Intersection finds commonality, while difference finds unique elements in one set compared to another.
The Intersection of A and B (A ∩ B) consists of elements that belong to both A and B. If A ∩ B = φ, the sets are called 'Disjoint Sets', meaning they have no elements in common.
The Difference of sets A and B (A – B) is the set of elements which belong to A but not to B. Note that A – B is generally not equal to B – A. For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A – B = {1} and B – A = {4}.
Finding Intersection
A = {3, 5, 7, 9}, B = {7, 9, 11, 13}
The elements common to both sets are 7 and 9. Therefore, A ∩ B = {7, 9}.
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If A = {a, b, c} and B = {d, e, f}, what is A ∩ B?
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07 · Explore
Complement of a Set and De Morgan's Laws
The complement of a set refers to everything in the universal set that is not in the specified set.
Let U be the universal set and A be a subset of U. The complement of A (denoted as A') is the set of all elements of U which are not in A. Mathematically, A' = U – A.
De Morgan's Laws are two powerful properties: 1) The complement of the union is the intersection of the complements: (A ∪ B)' = A' ∩ B'. 2) The complement of the intersection is the union of the complements: (A ∩ B)' = A' ∪ B'.
Other properties include the Complement Laws: (i) A ∪ A' = U and (ii) A ∩ A' = φ. Additionally, the Law of Double Complementation states (A')' = A, and the laws of empty and universal sets are φ' = U and U' = φ.
Calculating Complement
U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, A = {2, 4, 6, 8}
A' contains all elements in U that are not in A. A' = {1, 3, 5, 7, 9}.
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Question
What is A ∩ A'?
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Sign inNew here? Sign up freeNCERT reference: chapter PDF pages 18, 19, 20.