1. Presented By
TOUSIF AZIM
University Roll Number- 31140523051
Student of 1st Year, 1st Semester, Bachelor of Science (B.sc) Data Science,
Gurunanak Institute of Technology,
157/F, Nilgunj Road, Panihati, Kolkata-700114, West Bengal.
Gurunanak Institute of Technology,
Department of Data Science 1
04-01-2024
2. • Introduction
• Representation of Sets
• Subsets and Supersets
• Universal and Null Sets
• Basic Operations on Sets
• Laws of Set Algebra
• Conclusion
• References
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3. Sets, in mathematics, are an organized collection of objects
and can be represented in set-builder form or roster form.
Usually, sets are represented in curly braces {}, for example,
A = {1,2,3,4} is a set. Also, check the set symbols here. It
was developed to describe the collection of objects. You have
already learned about the classification of sets here. The set
theory defines the different types of sets, symbols and
operations performed.
Sets are represented as a collection of well-defined objects or
elements and it does not change from person to person. A set
is represented by a capital letter. The number of elements in
the finite set is known as the cardinal number of a set.
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4. • Statement Form:- In statement form, the well-defined descriptions of a member of a set are
written and enclosed in the curly brackets.
For example, the set of even numbers less than 15.
In statement form, it can be written as {even numbers less than 15}.
• Roster Form: Imagine listing all the members of your favorite band in a neat line. That's
roster form in action! It involves writing the elements within curly braces, separated by
commas. For example, the set of primary colors can be written as:
{red, blue, yellow}
Simple and intuitive: Easy to understand and write, especially for small sets.
Repetition isn't allowed: Remember, each element gets only one spot in the box!
Order doesn't matter: {red, green, blue} is the same as {green, red, blue}.
In Roster form, all the elements of a set are listed.
For example, the set of natural numbers less than 5.
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….
Natural Number less than 5 = 1, 2, 3, 4
Therefore, the set is N = { 1, 2, 3, 4 }
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5. • Set-Builder Form: Think of a recipe: you specify the ingredients and their properties to create a dish.
Set-builder form works similarly. We describe the characteristic property that all elements in the set
must share, enclosed within curly braces. For example, the set of even numbers less than 10 can be
written as: {n | n is even, n < 10}
Concise for larger sets: You don't need to list every element explicitly, making it efficient for infinite
sets.
Focuses on the defining property: Clarifies what makes an element belong to the set.
Requires clear understanding of the property: Make sure your description accurately captures the
intended elements.
The general form is, A = { x : property }
Example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}
Also, Venn Diagrams are the simple and best way for visualized representation of sets.
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6. Imagine you have a treasure chest overflowing with precious gems. Inside it, you keep a special
velvet pouch where you store your most dazzling diamonds. This hidden collection represents a
special relationship between sets – that of a subset and its superset.
Subsets: Think of a subset as a smaller box nestled inside a larger one. All the elements of the
smaller box (the subset) must also be found in the larger box (the superset). It's like a secret club
within a bigger group, sharing some members but not all.
For example, the set of vowels {a, e, i, o, u} is a subset of the alphabet {a, b, c, ..., x, y, z}.
Every vowel is also a letter, but not every letter is a vowel.
A set ‘A’ is said to be a subset of B if every element of A is also an element of B, denoted as A
⊆ B. Even the null set is considered to be the subset of another set. In general, a subset is a part
of another set.
Example: A = {1,2,3}
Then {1,2} ⊆ A.
Similarly, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.
Note: The set is also a subset of itself.
If A is not a subset of B, then it is denoted as A⊄B.
Proper Subset- If A ⊆ B and A ≠ B, then A is called the proper subset of B and it can be written
as A⊂B.
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7. Example: If A = {2,5,7} is a subset of B = {2,5,7} then it is not a proper subset of B =
{2,5,7}
But, A = {2,5} is a subset of B = {2,5,7} and is a proper subset also.
Supersets: The larger box, the superset, encompasses all the members of the smaller
box (the subset) and potentially more. It's like a family where everyone in the younger
generation belongs to the older generation, but the family might also include
grandparents, aunts, and uncles.
In our example, the alphabet is a superset of the vowels. It contains all the vowels and
then some.
Set A is said to be the superset of B if all the elements of set B are the elements of set
A. It is represented as A ⊃ B.
For example, if set A = {1, 2, 3, 4} and set B = {1, 3, 4}, then set A is the superset of
B.
Here are some key points to remember:
• A set can be a subset of multiple supersets. The set of even numbers is a
subset of both the integers and the positive numbers.
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8. • A set can have multiple subsets. The set of even numbers
has many subsets, including {2}, {4, 6}, and {2, 4, 6, 8, ...}.
• Every set is a subset of itself and a superset of the empty
set. This might seem strange, but it just means "everything is
inside itself" and "nothing is inside an empty box."
Understanding subsets and supersets is crucial for various
mathematical operations and concepts. They help us:
• Compare sets and analyze their relationships.
• Perform set operations like union and intersection with
greater clarity.
• Solve problems involving inclusion and exclusion of
elements.
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9. In the ever-expanding realm of sets, we stumble upon two special entities that hold
crucial roles – the universal set and the null set. They act as the alpha and omega, the
ultimate container and the void, setting the stage for defining and comparing all other
sets within their domain.
• Universal Set (U): Imagine a giant treasure chest, encompassing all possible
elements within a specific context. This all-encompassing box is the universal set. It
acts as the reference point, the ultimate superset from which all other sets derive
their meaning.
For example, if we're dealing with the set of planets in our solar system, the universal
set could be "all celestial bodies within our solar system." This includes planets, moons,
asteroids, comets, and everything else under the cosmic sun.
• Null Set (∅): Now, picture an empty box, devoid of any element. This vacuous entity
is the null set, sometimes denoted by {} or ϕ. It represents the absence of any
element within a specific context.
Going back to our solar system example, the null set would be the set of "unicorns
found in our solar system." Since unicorns are mythical creatures, there are none within
our solar system, making the null set the perfect representation of this emptiness.
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10. Here are some key points to remember:
• Every element in a set must belong to the universal set. If something isn't
in the big box, it doesn't exist in that context.
• The null set has no elements. It's like an empty stage, waiting for the
actors to arrive.
• The null set is a subset of every set. Even though it has nothing inside, it
still fits snugly within any bigger box.
• Understanding these special sets is crucial for several reasons:
• They provide a reference point for comparing sets and analyzing their
relationships.
• They enable us to define special types of sets like complements and power
sets.
• They offer valuable tools for solving problems involving inclusion,
exclusion, and emptiness.
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11. In set theory, the operations of the sets are carried when two or more sets combine to form a single set under some
of the given conditions. The basic operations on sets are:
• Union of sets
• Intersection of sets
• A complement of a set
• Cartesian product of sets.
• Set difference
Basically, we work more on union and intersection of sets operations, using Venn diagrams.
• Union of Sets-If set A and set B are two sets, then A union B is the set that contains all the elements of set A and
set B. It is denoted as A ∪ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A union B is:
A ∪ B = {1,2,3,4,5,6}
• Intersection of Sets
If set A and set B are two sets, then A intersection B is the set that contains only the common elements between set A
and set B. It is denoted as A ∩ B.
Example: Set A = {1,2,3} and B = {4,5,6}, then A intersection B is:
A ∩ B = { } or Ø
Since A and B do not have any elements in common, so their intersection will give null set.
Complement of Sets:- The complement of any set, say P, is the set of all elements in the universal set that are not in
set P. It is denoted by P’.
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12. Properties of Complement sets
1. P ∪ P′ = U
2. P ∩ P′ = Φ
3. Law of double complement : (P′ )′ = P
4. Laws of empty/null set(Φ) and universal set(U), Φ′ = U and U′ = Φ.
• Cartesian Product of sets:- If set A and set B are two sets then the cartesian product of set A and set
B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is
denoted by A × B.
We can represent it in set-builder form, such as:
A × B = {(a, b) : a ∈ A and b ∈ B}
Example: set A = {1,2,3} and set B = {Bat, Ball}, then;
A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}
• Difference of Sets:-
If set A and set B are two sets, then set A difference set B is a set which has elements of A but no
elements of B. It is denoted as A – B.
Example: A = {1,2,3} and B = {2,3,4}
A – B = {1}
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13. In the realm of sets, just as numbers have arithmetic laws, there exists a set of guidelines
that govern how sets interact with each other. These are known as the Laws of Set Algebra.
Think of them as the rules of a magical game, ensuring order and consistency as you
combine, rearrange, and explore the relationships between sets.
Here are some fundamental laws to guide your journey:
1. Idempotent Laws:
• A union itself is A. (A ∪ A = A)
• An intersection of a set with itself is the set itself. (A ∩ A = A)
2. Commutative Laws:
• The order of sets in a union doesn't matter. (A ∪ B = B ∪ A)
• The order of sets in an intersection doesn't matter. (A ∩ B = B ∩ A)
3. Associative Laws:
• How you group sets in a union doesn't change the result. (A ∪ (B ∪ C) = (A ∪ B)
∪ C)
• How you group sets in an intersection doesn't change the result. (A ∩ (B ∩ C) =
(A ∩ B) ∩ C)
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14. 4. Distributive Laws:
• Union distributes over intersection. (A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C))
• Intersection distributes over union. (A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C))
5. Identity Laws:
• The empty set is the identity for union (like 0 for addition). (A ∪ ∅ = A)
• The universal set is the identity for intersection (like 1 for multiplication). (A ∩ U =
A)
6. Complement Laws:
• A double complement is the original set. (A ∪ A' = U, A ∩ A' = ∅)
• De Morgan's Laws: The complement of a union is the intersection of complements,
and vice versa. (A ∪ B)' = A' ∩ B', (A ∩ B)' = A' ∪ B'
Mastering these laws unlocks a world of possibilities:
• Simplify complex set expressions.
• Prove set-related theorems and conjectures.
• Solve problems involving sets in various fields, including mathematics, computer
science, logic, and more.
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15. After Studying Set thus we can conclude that Sets, in their elegant simplicity, offer a versatile
language for organizing, exploring, and manipulating collections of objects. Their influence
extends far beyond the realm of mathematics, shaping numerous fields and unlocking new
ways of understanding the world. Sets provide the building blocks for more complex
mathematical structures, such as numbers, functions, relations, and more, forming a
cornerstone of modern mathematics. The language of sets fosters precise thinking and logical
reasoning, enabling us to analyze relationships, break down problems, and formulate solutions
in a systematic manner. Sets allow us to move beyond concrete examples and capture abstract
concepts, leading to broader insights and generalizations within and beyond mathematics. Set
theory finds applications in diverse fields, including: Computer Science in Data structures,
databases, algorithms, Engineering in Circuit design, optimization, control systems, Logic in
Formal reasoning, proof theory, Statistics in Probability, sampling techniques, Linguistics in
Language analysis, semantics, Philosophy in Set theory itself has philosophical implications.
Thus sets stand as a testament to the power of clarity and abstraction in mathematical thought.
Their versatility and impact continue to shape our understanding of the world and fuel
advancements across a multitude of disciplines. As we continue to explore the frontiers of
knowledge, sets will remain an indispensable tool, guiding our journeys towards deeper
insights and innovative solutions.
04-01-2024
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