Introduction to Set Theory

Introduction to Set Theory

• The formal use of "set" in mathematics was established by George Cantor in 1879.

Language and Notation of Sets

• If S is a set, then the notation ( x \in S ) indicates that x is an element of S, while ( x \notin S ) signifies that x is not an element of S.
• Set-roster notation lists all elements within braces, e.g., ( {1, 2, 3} ).
• To describe large or infinite sets:
  • ( {1, 2, 3, \ldots, 100} ) symbolizes integers from 1 to 100.
  • ( {1, 2, 3, \ldots} ) represents all positive integers.

Examples of Set-Roster Notation

• Sets A, B, and C all contain the same elements: 1, 2, and 3, even if represented differently.
• ( {0} ) is a set containing one element (0), whereas 0 alone is a number, not a set.
• The set ( {1, {1}} ) contains two elements: the number 1 and the set whose only element is 1.

Special Sets of Numbers

• The symbols commonly used in set notation include:
  • ( \mathbb{R} ): Set of all real numbers.
  • ( \mathbb{Z} ): Set of all integers.
  • ( \mathbb{Q} ): Set of all rational numbers (quotients of integers).

Set-Builder Notation

• A set can be defined by a property ( P(x) ) which elements satisfy, noted as ( {x \in S \mid P(x)} ).

Examples of Sets Using Set-Builder Notation

• A set representing all real numbers between -2 and 5 is ( {x \in \mathbb{R} \mid -2 < x < 5} ).
• A set of integers from -1 to 5 is represented as ( {x \in \mathbb{Z} \mid -1 \leq x < 6} ).

Subsets

• A is a subset of B (written ( A \subseteq B )) if every element of A is also in B.
• If at least one element of A is not in B, then A is not a subset.
• A is a proper subset of B if all elements of A are in B, and there exists at least one element in B not in A.

Examples of Subset Relationships

• Assessing sets A, B, and C:
  • ( B \subseteq A ): False, since 0 is not a positive integer.
  • ( C ) is a proper subset of ( A ): True, as each element of C is in A, but not vice versa.
  • Elements common in C and B: True.

Distinction Between Element and Subset Notation

• ( 2 \in {1, 2, 3} ) confirms 2 is an element of the set.
• ( 2 \subseteq {1, 2, 3} ) is incorrect, as 2 is not a set but an element.
• ( {2} \subseteq {1, 2, 3} ) is correct, as {2} is a set containing 2, which is part of the larger set.