Discrete Structures - Sets and Subsets Practice Flashcards

Vocabulary practice flashcards covering basic set theory, notation, subsets, and power sets based on the CSCI 1900 lecture notes.

Set

Any well-defined collection of objects.

Elements (Members)

The objects contained in a set.

Well-defined

The condition where it is possible to decide if a given object belongs to a collection or not.

$$Æ$$

The label for the empty set, where $Æ = \{ \}$.

$$Î$$

Notation meaning "is an element of," shaped like an "E" for element.

$$Ï$$

Notation meaning "is not an element of."

$$A = \{a \mid a \text{ is ...}\}$$

Notation representing "the set $A$ is comprised of elements $a$ where $a$ satisfies specified properties."

$$Z^+$$

The set of all $x$ such that $x$ is a positive integer.

$$N$$

The set of all $x$ such that $x$ is a positive integer or zero.

$$Z$$

The set of all $x$ such that $x$ is an integer.

$$Q$$

The set of rational numbers that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

$$R$$

The set of all $x$ such that $x$ is a real number.

Subset ($A \text{ Í } B$)

A condition where every element of $A$ is also an element of $B$, meaning for every $x$, $x \in A$ implies $x \in B$. Or, $A$ is "contained in" $B$.

$$Í$$

The notation for "is contained in," shaped like a "C" for contained.

Venn Diagrams

Named after British logician John Venn, these are graphical depictions of the relationship between sets.

Subset Transitive Property

The theorem stating that if $A \text{ Í } B$ and $B \text{ Í } C$, then $A \text{ Í } C$.

Reflexive Subset Property

The theorem stating that if $A$ is any set, then $A \text{ Í } A$ (every set is a subset of itself).

$$Æ \text{ Í } A$$

A theorem stating that the statement "the empty set is a subset of $A$" is always true for any set $A$, because $Æ$ contains no elements.

Set Equality

The condition where if $A \text{ Í } B$ and $B \text{ Í } A$, then $A = B$.

Universal Set ($U$)

The set containing all objects for which the discussion is meaningful; denoted with a rectangle in a Venn diagram.

Finite Set

A set $A$ that has $n$ distinct elements, where $n \in N$.

Cardinality

The number of distinct elements in a set $A$, denoted by $|A|$.

Infinite Set

A set that is not finite.

Power Set ($P(A)$)

The set of all subsets of $A$, including the empty set ($Æ$).

Cardinality of $P(A)$

Calculated by the formula $2^n$, where $n$ is the cardinality of set $A$.