🎯 COMPLETE SETS MATHEMATICS
All Definitions, Properties & Formulas
📌 BASIC DEFINITIONS
Set
A well defined collection of objects.
Objects, elements and members of set are synonyms terms.
Notation
Sets: \( A, B, C, X, Y, Z \) etc.
Elements: \( a, b, c, x, y, z \) etc.
Tableau Form
Elements are listed, separated by commas and enclosed within curly brackets
Example: \( \{a, e, i, o, u\} \) set of vowels
Set Builder Form
All elements possess a single common property
Example: \( \{x : x \in X \text{ is a vowel in English alphabet}\} \)
🔢 TYPES OF SETS
Cardinal Number
Number of elements of a Set \( A \) is called cardinal number
Notation: \( n(A) \)
Empty Set
A set which does not contain any element is called the empty set or null set or void set
Notation: \( \emptyset \)
Finite Set
A set which is empty or consists of a definite number of elements
Example: \( \{1, 2, 3, 4, 5\} \)
Infinite Set
A set which is not empty \( \emptyset \) consists of an indefinite number of elements
Example: \( \{1, 2, 3, \ldots\} \)
Equal Sets
Two sets \( A \) and \( B \) are equal if they have exactly the same elements
Notation: \( A = B \) or \( A \neq B \)
Singleton Set
If a set \( A \) has only one element, we call it singleton set
Example: \( \{a\} \)
🔗 SET RELATIONSHIPS
Subset
A set \( A \) is subset of \( B \) if every element of \( A \) is also in \( B \)
Notation: \( A \subseteq B \) if \( a \in A \Rightarrow a \in B \)
Proper Subset
If \( A \subset B \) and \( A \neq B \), then \( A \) is proper subset of \( B \)
\( B \) is called superset of \( A \)
Power Set
The collection of all subsets of a set \( A \)
Notation: \( P(A) \)
Universal Set
A set that contains all sets in a given context
Notation: \( U \)
Note: \( N \subset Z \subset Q, R, N^+ \) – Subsets of real numbers
📏 INTERVALS AS SUBSETS OF \( \mathbb{R} \)
\( (a, b) \)
Set Builder: \( \{ x : a < x < b \} \)
Open interval, does not contain endpoints
\( [a, b] \)
Set Builder: \( \{ x : a \leq x \leq b \} \)
Closed interval, contains endpoints
\( [a, b) \)
Set Builder: \( \{ x : a \leq x < b \} \)
Includes \( a \), excludes \( b \)
\( (a, b] \)
Set Builder: \( \{ x : a < x \leq b \} \)
Excludes \( a \), includes \( b \)
Length of any interval: \( (b-a) \) is the length of intervals
∪ UNION OF SETS
Definition
The union of \( A \) and \( B \) consists of all elements of \( A \) and \( B \), common elements taken only once
\( A \cup B = \{ x : x \in A \text{ or } x \in B \} \)
1. Commutative Law
\( A \cup B = B \cup A \)
2. Associative Law
\( (A \cup B) \cup C = A \cup (B \cup C) \)
3. Distributive Law
\( (A \cup B) \cap C = (A \cap C) \cup (B \cap C) \)
4. Identity Law
\( A \cup (\emptyset) = A \)
\( \emptyset \) is identity of \( \cup \)
5. Division Law
\( A \cap (\emptyset) = A \)
\( \emptyset \) is division of \( \cup \)
∩ INTERSECTION OF SETS
Definition
The intersection of \( A \) and \( B \) is the set of all elements common to both \( A \) and \( B \)
\( A \cap B = \{ x : x \in A \text{ and } x \in B \} \)
Commutative Law
\( A \cap B = B \cap A \)
Associative Law
\( (A \cap B) \cap C = A \cap (B \cap C) \)
Idempotent Law
\( A \cap A = A \)
Identity Laws
\( \phi \cap A = \phi \)
\( U \cap A = A \)
Distributive Law
\( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)
− DIFFERENCE & COMPLEMENT
Difference of Sets
Elements which belong to \( A \) but not to \( B \)
\( A – B = \{ x : x \in A \text{ and } x \notin B \} \)
Note: \( A – B \neq B – A \)
Complement of a Set
Let \( U \) be universal set and \( A \subset U \)
Complement of \( A \) is elements of \( U \) not in \( A \)
\( A’ = \{ x : x \in U \text{ and } x \notin A \} \)
Complement & De Morgan’s Laws
Complement Laws
\( A \cup A’ = U \)
\( A \cap A’ = \emptyset \)
\( (A’)’ = A \)
De Morgan’s Laws
\( (A \cup B)’ = A’ \cap B’ \)
\( (A \cap B)’ = A’ \cup B’ \)
Other Laws
\( \phi’ = U \)
\( U’ = \phi \)
📊 FORMULAS
Cardinality Formulas
For two sets:
\( n(A \cup B) = n(A) + n(B) – n(A \cap B) \)
For three sets:
\( n(A \cup B \cup C) = n(A) + n(B) + n(C) \)
\( – n(A \cap B) – n(B \cap C) – n(C \cap A) \)
\( + n(A \cap B \cap C) \)
Number Systems
\( \mathbb{N} \): Natural Numbers \( \{1, 2, 3, \ldots\} \)
\( \mathbb{Z} \): Integers \( \{\ldots, -2, -1, 0, 1, 2, \ldots\} \)
\( \mathbb{Q} \): Rational Numbers
\( \mathbb{R} \): Real Numbers
\( \mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \)
Complete Sets Mathematics Notes
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