Set Theory

πŸ“˜ SETS

πŸ”Ή What is a Set?

Definition:
A set is a well-defined collection of distinct objects. These objects are called elements or members of the set. Sets are usually written inside curly brackets { }.

Real-life Example:
πŸ‘‰ In a school, consider the set of students who play different games.

  • Set A = {students who play Cricket}
  • Set B = {students who play Football}
  • Set C = {students who play Chess}

We’ll use this school games example to explain all types of sets.

Two Ways to Represent a SetThere are two main ways to write down a set:

the Roster Form and the Set-Builder Form.

1. Roster Form (or Tabular Form)

In this form, you list all of the individual items in the set. You put them inside curly braces { } and separate each item with a comma.

The most important rule is that you do not write the same item more than once. Each element is unique in the list.

Examples:The set of vowels in English: {a, e, i, o, u}

The set of students who play cricket: {Rahul, Priya, Ankit, Riya}

2. Set-Builder Form

In this form, you describe the set by giving a rule or a condition that all its members share. Instead of listing items, you define what makes something a member.

The standard way to write it is: {x | condition on x}

This is read as: “The set of all x such that x meets the following condition.”

Examples:The set of vowels in English: {x | x is a vowel in the English alphabet}

The set of students who play cricket: {x | x is a student who plays cricket in the school}

πŸ”Ή Types of Sets

1. Empty Set (Null Set)

Definition: A set which contains no elements.
Example (School): Set of students who can fly like birds = Ο•

2. Singleton Set

Definition: A set that contains only one element.
Example (School): {Principal}

3. Finite Set

Definition: A set with a countable number of elements.
Example (School): {students in Class 10}

4. Infinite Set

Definition: A set with uncountably many elements.
Example (School): Set of natural numbers {1, 2, 3, …} (marks scored by students can be part of this idea).

5. Equal Sets

Definition: Two sets are equal if they have exactly the same elements.
Example (School):
A = {1, 2, 3}, B = {3, 2, 1} β†’ A = B

6. Equivalent Sets

Definition: Two sets are equivalent if they have the same number of elements, though not necessarily the same elements.
Example (School):
Set of {3 different games} and set of {3 classroom subjects} β†’ Equivalent sets

7. Subset

Definition: A set A is a subset of B if every element of A is also in B.
Example (School): Cricket players βŠ† All students in school

8. Proper Subset

Definition: A proper subset is a subset that is not equal to the set itself.
Example (School): {Girls in school} βŠ‚ {All students in school}

9. Power Set

Definition: The set of all subsets of a given set.
Example (School):
If A = {a, b}, then
P(A) = {Ο•, {a}, {b}, {a, b}}

10. Universal Set

Definition: The set that contains all elements under discussion.
Example (School): U = {all students in school}

11. Disjoint Sets

Definition: Two sets are disjoint if they have no elements in common.
Example (School):
A = {students who play Cricket},
B = {students who play Chess}
If no student plays both β†’ A ∩ B = Ο•

12. Overlapping Sets

Definition: Two sets are overlapping if they have at least one common element.
Example (School):
A = Cricket players, B = Football players
Some students play both β†’ A ∩ B β‰  Ο•

13. Complement of a Set

Definition: All elements of the universal set which are not in A.
Example (School):
U = {all students}, A = {Cricket players}
Aβ€² = {students who do not play Cricket}

14. Union of Sets

Definition: The union of two sets is the set containing all elements of both sets.
Example (School):
A = Cricket players, B = Football players
A βˆͺ B = {students who play Cricket or Football (or both)}

15. Intersection of Sets

Definition: The intersection of two sets is the set containing only the common elements.
Example (School):
A = Cricket players, B = Football players
A ∩ B = {students who play both Cricket and Football}

πŸ”Ή Final Connected Picture (School Example)

  • Universal Set (U): All students in the school
  • Subset: Cricket players βŠ† U
  • Proper Subset: Girls βŠ‚ U
  • Disjoint Sets: Cricket players ∩ Chess players = Ο•
  • Overlapping Sets: Cricket players ∩ Football players β‰  Ο•
  • Complement: Non-cricket players = U – A
  • Union: Cricket βˆͺ Football = Students who play at least one

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Essential Properties of Sets (With Formulas & Examples)

  1. Commutative Laws
    Definition: The order in which you combine sets does not matter.
    Union: A βˆͺ B = B βˆͺ A
    Intersection: A ∩ B = B ∩ A

Example (School Clubs):
Cricket βˆͺ Football is the same group as Football βˆͺ Cricket (all players).
Cricket ∩ Football is the same group as Football ∩ Cricket (players who play both).

  1. Associative Laws
    Definition: The grouping of sets does not affect the result of union or intersection.
    Union: (A βˆͺ B) βˆͺ C = A βˆͺ (B βˆͺ C)
    Intersection: (A ∩ B) ∩ C = A ∩ (B ∩ C)

Example (School Clubs):
(Cricket βˆͺ Football) βˆͺ Chess is the same as Cricket βˆͺ (Football βˆͺ Chess) (all players from any club).
(Cricket ∩ Football) ∩ Chess is the same as Cricket ∩ (Football ∩ Chess) (players who play all three).

  1. Distributive Laws
    Definition: Union distributes over intersection, and intersection distributes over union.
    Union over Intersection: A βˆͺ (B ∩ C) = (A βˆͺ B) ∩ (A βˆͺ C)
    Intersection over Union: A ∩ (B βˆͺ C) = (A ∩ B) βˆͺ (A ∩ C)

Example (School Clubs):
Left side: Students in Cricket OR in (Football AND Chess).
Right side: Students are (in Cricket or Football) AND (in Cricket or Chess).
Both descriptions define the same group of students.

  1. Idempotent Laws
    Definition: The union or intersection of a set with itself yields the original set.
    Union: A βˆͺ A = A
    Intersection: A ∩ A = A

Example (School Clubs):
Cricket βˆͺ Cricket = Cricket (The list of cricket players remains unchanged).
Cricket ∩ Cricket = Cricket (The players common to the cricket team and itself is the whole team).

  1. Identity Laws
    Definition: The union with the empty set (βˆ…) or intersection with the universal set (U) leaves a set unchanged.
    Union with βˆ…: A βˆͺ βˆ… = A
    Intersection with U: A ∩ U = A

Example (School Clubs):
Cricket βˆͺ {} = Cricket (Adding no one to the team doesn’t change it).
Cricket ∩ {All Students} = Cricket (The students who are both “all students” and “cricket players” are just the cricket players).

  1. Domination (Null) Laws
    Definition: The union with the universal set absorbs the set, and the intersection with the empty set nullifies it.
    Union with U: A βˆͺ U = U
    Intersection with βˆ…: A ∩ βˆ… = βˆ…

Example (School Clubs):
Cricket βˆͺ {All Students} = {All Students} (Combining a club with everyone results in everyone).
Cricket ∩ {} = {} (There are no students who are both cricket players and no one).

  1. Complement Laws
    Definition: A set combined with its complement gives the universal set or the empty set.
    Union with Complement: A βˆͺ A’ = U
    Intersection with Complement: A ∩ A’ = βˆ…

Example (School Clubs):
Cricket βˆͺ Non-Cricket Players = All Students
Cricket ∩ Non-Cricket Players = {} (No one can both play and not play cricket).

  1. Double Complement (Involution) Law
    Definition: The complement of a complement is the original set.
    Formula: (A’)’ = A

Example (School Clubs):
The complement of “Non-Cricket Players” is “Cricket Players”.

  1. De Morgan’s Laws
    Definition: Rules for finding the complement of unions and intersections.
    Complement of a Union: (A βˆͺ B)’ = A’ ∩ B’
    Complement of an Intersection: (A ∩ B)’ = A’ βˆͺ B’

Example (School Clubs):
First Law: Students who are not in (Cricket OR Football) are the same as students in (Non-Cricket AND Non-Football).
Second Law: Students who are not in (Cricket AND Football) are the same as students in (Non-Cricket OR Non-Football).

  1. Absorption Laws
    Definition: One set “absorbs” another when combined in a specific way.
    Union absorbs Intersection: A βˆͺ (A ∩ B) = A
    Intersection absorbs Union: A ∩ (A βˆͺ B) = A

Example (School Clubs):
Cricket βˆͺ (Cricket ∩ Football) = Cricket (The cricket team, plus the players who are on both teams, is just the cricket team).
Cricket ∩ (Cricket βˆͺ Football) = Cricket (The players common to the cricket team and the combined (cricket+football) group are just the cricket players).

  1. Set Difference
    Definition: The elements that are in set A but not in set B.
    Formula: A – B = {x | x ∈ A and x βˆ‰ B}

Example (School Clubs):
Cricket – Football = Students who play only Cricket (they are in Cricket but not in Football).

  1. Symmetric Difference
    Definition: The elements that are in either set A or set B, but not in both.
    Formula: A Ξ” B = (A – B) βˆͺ (B – A)

Example (School Clubs):
Cricket Ξ” Football = Students who play either Cricket or Football, but not both.