Study sheet: Set Theory Fundamentals

📋 Course Outline

1. Sets and Their Elements
2. Finite and Infinite Sets
3. Subsets and Universal Sets
4. Set Operations
5. Set Algebra and Relations
6. Cardinalities and Counting
7. Translating Set Descriptions

📖 1. Sets and Their Elements

🔑 Key Concepts & Definitions

• Set : Georg Cantor — a collection of clearly distinguished objects considered as a whole.
• Element membership : The notation $a \in A$ means that a belongs to set A, whereas $a \notin A$ means that a does not belong to set A.
• Set equality : Two sets are equal when they contain exactly the same elements, regardless of the order in which those elements are written.

📝 Essential Points

📌 A set can be specified in roster form by listing its elements, or in descriptive form by stating the condition satisfied by its elements.

💡 Memory Hook

Roster form lists elements; descriptive form states a membership condition.

📖 2. Finite and Infinite Sets

★ Must-know

📌 A finite set has a listing that eventually stops, whereas an infinite set has a listing that never stops.

Further detail

• The natural divisors of 6 form the finite set $T_6=\{1{,}2{,}3{,}6\}$, while the natural multiples of 6 form the infinite set $V_6=\{6{,}12{,}18,\dots\}$.

💡 Memory Hook

Finite sets stop being listed; infinite sets continue indefinitely.

📖 3. Subsets and Universal Sets

🔑 Key Concepts & Definitions

• Subset : The notation $A\subset B$ means that every element of A also belongs to B, and B is then a superset of A.
• Empty set : The empty set contains no elements and is a subset of every set.
• Universal set : The universal set G contains all objects allowed in the set discussion, and the sets under consideration are subsets of G.

📝 Essential Points

• A set with n elements has $2^n$ subsets.

💡 Memory Hook

Picture every subset as a smaller circle inside the universal set G.

📖 4. Set Operations

🔑 Key Concepts & Definitions

• Intersection : The intersection $A\cap B$ consists of the elements that belong to both A and B.
• Union : The union $A\cup B$ consists of the elements that belong to A, to B, or to both.
• Set difference : The difference $A\setminus B$ consists of the elements that belong to A but not to B.
• Complement : The complement of A consists of all elements of the universal set G that do not belong to A.

💡 Memory Hook

Intersection means both, union means at least one, and difference means the first but not the second.

📖 5. Set Algebra and Relations

★ Must-know

📌 The identities $A\cap A=A$, $A\cup A=A$, and $A\setminus A=\varnothing$ simplify repeated set operations.

Further detail

📌 The identities $A\cap(A\cup B)=A$ and $A\cup(A\setminus B)=A$ express absorption in set algebra.

📌 Two sets are disjoint when their intersection is empty, written $A\cap B=\varnothing$.

💡 Memory Hook

Simplify with intersection, union, difference, and complement.

📖 6. Cardinalities and Counting

📝 Essential Points

📐 Formula — For finite sets, the cardinality of a union satisfies $|A\cup B|=|A|+|B|-|A\cap B|$.

📐 Formula — The cardinality of a difference satisfies $|A\setminus B|=|A|-|A\cap B|$.

• 🔄 The counting process is:
  1. count the first set
  2. count the second set
  3. subtract the overlap
  4. obtain the union

💡 Memory Hook

Overlap reduces the union: counting A and B together requires subtracting their intersection.

📖 7. Translating Set Descriptions

★ Must-know

• A set can be converted from descriptive form to roster form by identifying every object that satisfies the stated condition and listing those objects.

Further detail

• A set can be converted from roster form to descriptive form by identifying the common property of all listed elements and expressing that property as a condition.

• The set $\{1{,}3{,}7{,}21\}$ can be described as the set of natural divisors of 21.

• The set $\{1{,}4{,}9{,}16{,}25,\dots\}$ can be described as the set of natural square numbers.

💡 Memory Hook

Move between words, roster notation, descriptive notation, and set terms.

📊 Synthesis Tables

Basic Set Operations

Operation	Meaning	Notation
Intersection	In both A and B	$A\cap B$
Union	In A, B, or both	$A\cup B$
Difference	In A but not B	$A\setminus B$
Complement	In G but not A	$G\setminus A$