Unit 1: Sets
Mathematics notes · PTU syllabus (UGCC2501)
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Unit summary
Sets are the language of modern mathematics and of computing. A set is simply a well-defined collection of distinct objects, and almost every later idea in this course (relations, functions, databases, logic) is built on it.
This unit covers how to write a set, the main types of sets, and the four operations you will use constantly: union, intersection, difference and complement.
After this unit you can
- Write any set in roster form and in set-builder form
- Identify null, singleton, finite, infinite, equal, equivalent and disjoint sets
- Find subsets, proper subsets and the power set of a set
- Perform union, intersection, difference, symmetric difference and complement, and draw them as Venn diagrams
PTU syllabus topics
- Introduction
- representation of sets (roster method, set builder method)
- types of sets (null, singleton, finite, infinite, equal, equivalent, disjoint, subset, proper subset, power set, universal set)
- set operations (union, intersection, difference, symmetric difference)
- complement of a set
Contains
Every element in A or B or both
Only elements in both A and B
Example: A={1,2,3}, B={3,4}
{1,2,3,4}
{3}
Difference A − B
Elements in A but not in B: {1,2}
Symmetric difference: {1,2,4}
Topic 1
Introduction to sets
A set is a well-defined collection of distinct objects. "Well-defined" means that for any object we can say clearly whether it belongs to the set or not. "The vowels of English" is a set; "the good students of a class" is not, because "good" is a matter of opinion. The objects in a set are called its elements or members. We write sets with capital letters (A, B, C) and elements with small letters. If a is an element of A, we write a ∈ A; if not, a ∉ A.
- Order does not matter: {1, 2, 3} and {3, 1, 2} are the same set.
- Repetition does not matter: {1, 1, 2} is the same as {1, 2}.
- The number of elements in a finite set A is its cardinality, written n(A).
Example
A = {a, e, i, o, u}. Here e ∈ A, b ∉ A and n(A) = 5.
Topic 2
Representation of sets
There are two standard ways to write a set.
| Method | How it works | Example |
|---|---|---|
| Roster (tabular) form | List every element inside braces, separated by commas | A = {2, 4, 6, 8, 10} |
| Set-builder (rule) form | State the property shared by all elements | A = {x : x is an even natural number, x ≤ 10} |
Roster form is best for small sets. Set-builder form is best for large or infinite sets, where listing is impossible.
Example
The set of natural numbers less than 6: roster form {1, 2, 3, 4, 5}; set-builder form {x : x ∈ N, x < 6}.
Exam tip
In the exam, when asked to "convert" a set, always show both forms side by side and mention which is which.
Topic 3
Types of sets
Null (empty)
No elements: { } or φ
Singleton
Exactly one element: {7}
Finite
Countable number of elements
Infinite
Elements never end: N, Z
Equal
Exactly the same elements
Equivalent
Same number of elements
- Null (empty) set: contains no element, written φ or { }. Example: {x : x is a natural number less than 1}.
- Singleton set: has exactly one element, such as {0}. Note that {0} is not empty — it contains 0.
- Finite and infinite sets: a finite set can be counted to an end ({1, 2, 3}); an infinite set cannot (the set of all natural numbers).
- Equal sets: A = B when they have exactly the same elements, e.g. {1, 2, 3} = {3, 2, 1}.
- Equivalent sets: have the same number of elements, n(A) = n(B), even if the elements differ: {a, b, c} and {1, 2, 3}.
- Disjoint sets: have no element in common, so A ∩ B = φ.
- Universal set (U): the set containing all elements under discussion in a problem.
Exam tip
Every equal pair of sets is equivalent, but equivalent sets need not be equal. This is a favourite short question.
Topic 4
Subsets, proper subsets and power set
A is a subset of B (A ⊆ B) if every element of A is also in B. A is a proper subset of B (A ⊂ B) if A ⊆ B and A ≠ B.
- Every set is a subset of itself, and φ is a subset of every set.
- If a set has n elements, it has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
- The power set P(A) is the set of all subsets of A, so n(P(A)) = 2ⁿ.
Example
A = {1, 2}. Subsets: φ, {1}, {2}, {1, 2}. So P(A) = {φ, {1}, {2}, {1, 2}} and n(P(A)) = 2² = 4.
Topic 5
Operations on sets
- Union A ∪ B
- Elements in A or B or both
- Intersection A ∩ B
- Elements common to A and B
- Difference A − B
- Elements in A but not in B
- Symmetric difference A Δ B
- Elements in exactly one of A and B: (A − B) ∪ (B − A)
- Complement A'
- Elements of U not in A: U − A
Take U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3, 4} and B = {3, 4, 5, 6}.
| Operation | Result |
|---|---|
| A ∪ B | {1, 2, 3, 4, 5, 6} |
| A ∩ B | {3, 4} |
| A − B | {1, 2} |
| B − A | {5, 6} |
| A Δ B | {1, 2, 5, 6} |
| A' | {5, 6, 7, 8} |
Useful laws to remember:
- Commutative: A ∪ B = B ∪ A and A ∩ B = B ∩ A
- De Morgan's laws: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'
- Counting formula: n(A ∪ B) = n(A) + n(B) − n(A ∩ B)
Example
In a class of 60, 35 study Hindi, 30 study English and 15 study both. Students studying at least one language = 35 + 30 − 15 = 50.
Key terms
- Set
- A well-defined collection of distinct objects
- Cardinality
- The number of elements in a finite set, n(A)
- Power set
- The set of all subsets of a set; it has 2ⁿ elements
- Universal set
- The set of all elements under consideration
- Disjoint sets
- Sets with no common element
- Complement
- Elements of the universal set that are not in the given set
Quick revision
- Roster form lists elements; set-builder form states a rule.
- A set with n elements has 2ⁿ subsets and 2ⁿ − 1 proper subsets.
- Equal sets have the same elements; equivalent sets have the same number of elements.
- A Δ B = (A − B) ∪ (B − A).
- De Morgan: (A ∪ B)' = A' ∩ B' and (A ∩ B)' = A' ∪ B'.
Important exam questions
Practice questions written to the PTU exam pattern for this unit's syllabus: short answers (Section A style) and long answers (Sections B and C style).
Short-answer questions
- Q1.Define a set. Give one example of a collection that is not a set.
- Q2.Differentiate between equal and equivalent sets with examples.
- Q3.Write the set {x : x is a prime number less than 15} in roster form.
- Q4.What is a power set? Find the power set of {a, b, c}.
- Q5.Define the symmetric difference of two sets.
- Q6.State De Morgan's laws for sets.
Long-answer questions
- Q1.Explain the different types of sets with suitable examples.
- Q2.If U = {1, 2, …, 10}, A = {2, 4, 6, 8} and B = {1, 2, 3, 4, 5}, find A ∪ B, A ∩ B, A − B, B − A, A Δ B and A'. Verify De Morgan's laws.
- Q3.Explain the roster and set-builder methods of representing sets, and convert three sets of your choice from one form to the other.
- Q4.In a survey of 100 students, 60 like tea, 50 like coffee and 25 like both. How many like neither? Illustrate with a Venn diagram.
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