Set Theory

These flashcards cover the fundamental concepts of set theory as presented in the lecture notes.

Set

A collection of distinct elements or objects, denoted by capital letters.

Element

An object belonging to a set, usually denoted by small letters.

Subset

A set A is a subset of B if every element of A is also an element of B, denoted A ⊂ B.

Proper Subset

A set A is a proper subset of B (A ⊂ B and A ≠ B) if it contains some, but not all, elements of B.

Empty Set

The set having no elements, denoted by ∅ or { }.

Universal Set

The set containing all elements relevant to a particular discussion, denoted by U or E.

Relative Complement

The set of all elements in A that are not in B, denoted by A - B.

Union of Sets

The set containing all elements that are in A, in B, or in both, denoted A ∪ B.

Intersection of Sets

The set containing all elements that are common to both A and B, denoted A ∩ B.

Power Set

The set of all subsets of a set A, denoted P(A) or 2^A.

Finite Set

A set with a specific number of elements, denoted as m where m is a positive integer.

Infinite Set

A set with no finite number of elements, such as the set of all positive integers.

Disjoint Sets

Two sets whose intersection is the empty set, denoted A ∩ B = ∅.

De Morgan’s Laws

The laws that relate the complement of unions and intersections of sets.

Countable Set

A set that can be put in one-to-one correspondence with the natural numbers.

Nesting of Sets

A collection or class of sets that can contain other sets, such as B = {{1,2},{3},{1,2,3}}.

Cardinality

The number of elements in a set, denoted by n(A).

Venn Diagram

A diagram that shows all possible logical relations between a finite collection of sets.

Reflexive Property

For any set A, A is a subset of itself, A ⊂ A.

Antisymmetric Property

If A ⊂ B and B ⊂ A, then A = B.

Transitive Property

If A ⊂ B and B ⊂ C, then A ⊂ C.