Set Theory Practice Flashcards

A complete set of 50 vocabulary flashcards covering the basic concepts, operations, and number systems of set theory based on the lecture transcript.

Set

A collection of certain objects that are well-distinguished from each other.

Elements of a Set

The objects that make up a set.

Georg Cantor (1845–1918)

The German mathematician who is considered the founder of set theory.

Symbol $\in$

Represents belonging (an element is part of a set).

Symbol $\notin$

Represents not belonging to a set.

Set Notation

Sets are usually denoted by uppercase Latin letters such as $A$, $B$, or $M$.

Recording Elements

Elements are written inside braces $\{ \}$ and separated by commas or semicolons.

Empty Set ($\emptyset$)

A set that does not contain any elements.

Universal Set ($U$)

A set that contains all possible elements considered within a specific context.

Cardinal Number ($n(A)$)

The quantity of elements contained within a set.

Subset

Set $A$ is a subset of $B$ if every element of $A$ is also an element of $B$.

Symbols $\subset$ or $\subseteq$

Used to signify that one set is a subset of another.

Equal Sets ($A = B$)

When two sets have exactly the same elements, meaning $A \subseteq B$ and $B \subseteq A$.

Finite Set

A set whose number of elements can be expressed as a natural number or zero.

Infinite Set

A set whose number of elements cannot be expressed by a natural number.

Proper Subset

A subset that is not equal to the whole set and is not empty.

Empty Set Property

The empty set is a subset of any set, denoted as $\emptyset \subseteq A$ for every $A$.

Descriptive Method

Characterizing a set by describing the properties of its elements.

Enumeration Method

Directly naming all elements of a set within braces.

Number of Subsets

A set with $n$ elements contains $2^n$ subsets.

Union ($A \cup B$)

A set consisting of elements that belong to $A$, or $B$, or both simultaneously.

Intersection ($A \cap B$)

A set containing only the common elements of sets $A$ and $B$.

Difference ($A \setminus B$)

A set containing all elements of $A$ that do not belong to $B$.

Complement ($\overline{A}$ or $A'$)

All elements in the universal set that are not included in set $A$.

Symmetric Difference ($A \Delta B$)

The set $(A \setminus B) \cup (B \setminus A)$, consisting of elements that belong to only one of the sets.

Intersection of a Set and its Complement ($A \cap \overline{A}$)

Always results in the empty set ($\emptyset$).

Union of a Set and its Complement ($A \cup \overline{A}$)

Always results in the universal set ($U$).

Set Difference of a Set with Itself ($A \setminus A$)

Always results in the empty set ($\emptyset$).

Union with the Empty Set ($A \cup \emptyset$)

Results in the original set $A$.

Intersection with the Empty Set ($A \cap \emptyset$)

Results in the empty set ($\emptyset$).

Intersection with the Universal Set ($A \cap U$)

Results in the original set $A$.

First De Morgan's Law

Expressed as $(A \cup B)' = A' \cap B'$.

Second De Morgan's Law

Expressed as $(A \cap B)' = A' \cup B'$.

Commutativity of Union

The property stated as $A \cup B = B \cup A$.

Associativity of Intersection

The property stated as $A \cap (B \cap C) = (A \cap B) \cap C$.

Symbol $\mathbb{N}$

Represents the set of natural numbers $\{1, 2, 3, \dots\}$.

Symbol $\mathbb{Z}$

Represents the set of integers $\{\dots, -2, -1, 0, 1, 2, \dots\}$.

Symbol $\mathbb{Q}$

Represents the set of rational numbers, which are numbers that can be expressed as fractions.

Symbol $I$

Represents the set of irrational numbers, such as $\sqrt{2}$ and $\pi$.

Symbol $\mathbb{R}$

Represents the set of real numbers.

Hierarchy of Number Sets

The inclusion relationship defined as $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$.

Natural Number

A number used for counting objects.

Zero ($0$) in Natural Numbers

Standard natural numbers start from 1, though some definitions include 0 ($\mathbb{N}_0$).

Integer

A set composed of natural numbers, their opposites, and zero.

Rational Number General Form

The form $\frac{a}{b}$, where $a \in \mathbb{Z}$ and $b \in \mathbb{N}$.

Open Interval $(a, b)$

A set of numbers where $a < x < b$, excluding the endpoints.

Closed Interval $[a, b]$

A set of numbers where $a \leq x \leq b$, including the endpoints.

Half-open Interval $[a, b)$

A set of numbers where $a \leq x < b$.

Symbol $\infty$

Represents infinity.

Hilbert's Hotel Paradox

An illustration showing that an infinite set can accept new elements even when it is already full.