IN1004: Mathematics for Computing - Set Theory

50 vocabulary flashcards covering key definitions, symbols, subset relations, power sets, standard number systems, and enumerability from the IN1004 Set Theory lecture notes.

Set

A well-defined collection of objects that forms a cornerstone of mathematics.

Elements

The objects contained within a set that are said to be members of the set.

Well-Defined

A property of a set implying that one can determine whether an object belongs to the set, avoiding sets based on opinion.

Capital Letters (Set Theory)

Notation used to represent sets, such as $A$, $B$, and $C$.

Lowercase Letters (Set Theory)

Notation used to represent elements of a set, such as $x$, $y$, and $z$.

Symbol $\in$

Indicates set membership; $x \in A$ means that $x$ is an element of set $A$.

Symbol $\notin$

Indicates the negation of set membership; $y \notin A$ means that $y$ is not a member of set $A$.

Set Braces

Curly brackets $\{\}$ used to enclose the listed elements or rules defining a set.

Vertical Line $|$ (Set Notation)

A symbol used within set comprehension read as 'such that'.

Expression Before $|$ (Set Comprehension)

The component of set comprehension notation that generates elements subject to given constraints.

Expression After $|$ (Set Comprehension)

The component of set comprehension notation that constrains the elements of the set being described.

Set Comprehension

A notation (also called set abstraction) that defines a set by specifying constraints on its elements following a vertical line $|$.

Universe of Discourse

An agreed-upon default set (usually denoted by $U$) from which elements are selected to form sets.

Symbol $U$

Standard notation used to represent the universe or universe of discourse.

Ellipses $\dots$

Notation used in set listings to represent an infinite continuation of a sequence.

Finite Set

A set that contains a fixed, countable number of elements.

Infinite Set

A set that contains an unending sequence of elements without a finite bound.

Cardinality

A measure of the number of elements in a finite set $A$, denoted by $|A|$ or $n(A)$.

Symbol $|A|$

Notation representing the cardinality or size of a set $A$.

Subset ($\subseteq$)

Set $C$ is a subset of set $D$ (written $C \subseteq D$ or $D \supseteq C$) if every element of $C$ is an element of $D$.

Proper Subset ($\subset$)

Set $C$ is a proper subset of $D$ (written $C \subset D$ or $D \supset C$) if $D$ contains all elements of $C$ and at least one extra element not in $C$.

Set Equality

Two sets $C$ and $D$ are equal ($C = D$) when $C \subseteq D$ and $D \subseteq C$.

Order Relevance in Sets

The mathematical principle that the arrangement of elements inside a set does not change the set, e.g., $\{1,2,3\} = \{3,2,1\}$.

Repetition Relevance in Sets

The mathematical principle that duplicate listings of elements do not alter a set, e.g., $\{1,2,3\} = \{1,2,1,2,3\}$.

Negation of Subset Relation ($\not\subseteq$)

Written as $A \not\subseteq B$ if there is at least one element $x$ in the universe where $x \in A$ but $x \notin B$.

Subsets Theorem i

The property stating that every set is a subset of itself ($A \subseteq A$).

Subsets Theorem iii

The transitive property of subsets stating that if $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$.

Empty Set

The unique set containing no elements, also known as the null set.

Symbol $\emptyset$

The standard symbol denoting the empty set or null set.

Cardinality of Empty Set

The size of the empty set, which is $|\emptyset| = 0$.

Distinction Between $\emptyset$ and $\{\emptyset\}$

$\emptyset$ contains zero elements, whereas $\{\emptyset\}$ is a set containing one element (the empty set).

Empty Set Subset Property

The fundamental property stating that the empty set is a subset of every set.

Power Set

The collection or set of all subsets of a set $A$, denoted by $P(A)$.

Symbol $P(A)$

Notation for the power set of set $A$.

Subset Count Formula

For any finite set $A$ with cardinality $|A| = n \ge 0$, the set $A$ has $2^n$ subsets.

Cardinality of Power Set

The size of a power set, given by $|P(A)| = 2^n$ where $n = |A|$.

Set $\mathbb{Z}$

The set of integers, defined as $\{0, 1, -1, 2, -2, 3, -3, \dots\}$.

Set $\mathbb{N}$

The set of natural numbers (non-negative integers), defined as $\{0, 1, 2, 3, \dots\}$.

Set $\mathbb{Z}^+$

The set of positive integers, defined as $\{1, 2, 3, \dots\}$.

Set $\mathbb{Q}$

The set of rational numbers, defined as $\left\{\frac{a}{b} \mid a,b \in \mathbb{Z}, b \neq 0\right\}$.

Set $\mathbb{Q}^+$

The set of positive rational numbers, defined as $\{r \mid r \in \mathbb{Q}, r > 0\}$.

Set $\mathbb{R}$

The set of real numbers, which includes rational numbers as well as irrational numbers like $\pi$, $e$, and $\sqrt{2}$.

Set $\mathbb{R}^+$

The set of positive real numbers.

Set $\mathbb{C}$

The set of complex numbers.

Number Systems Inclusion Chain

The subset relations among standard number systems: $\mathbb{N} \subseteq \mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}$.

Countable Set

A set (also called enumerable) whose elements can be ordered in a simple way so that they can be counted step-by-step.

Rearrangement of $\mathbb{Z}$ for Counting

The sequence $\{0, -1, 1, -2, 2, -3, 3, -4, 4, \dots\}$ used to establish a one-to-one mapping onto $\mathbb{N}$.

Cardinality Comparison of $\mathbb{N}$ and $\mathbb{Z}$

The sets $\mathbb{N}$ and $\mathbb{Z}$ are the same size (cardinality) because a one-to-one mapping can be constructed between them.

Cardinality Comparison of $\mathbb{N}$ and $\mathbb{R}$

The set $\mathbb{R}$ is not the same size as $\mathbb{N}$; it represents a larger infinity.

Java int Data Type Bounds

A primitive type using 32 bits for storage, which allows integers between $-2^{31}$ and $2^{31} - 1$.