Mathematics I - Sets

Vocabulary flashcards covering basic set definitions, subset relations, power sets, set operations, algebraic laws, and Cartesian products from Mathematics I.

Set

A collection of distinct objects or elements.

Element

An object contained in a set, denoted by lowercase letters (e.g., $a \in A$ indicates that $a$ is an element of set $A$, while $a \notin A$ means it is not).

Empty Set

The set that contains no elements, denoted by $\emptyset$. It is a subset of every set ($\emptyset \subseteq A$).

Subset

A set $A$ is a subset of $B$ if every element of $A$ is an element of $B$, denoted $A \subseteq B :\iff (\forall x : x \in A \implies x \in B)$.

Set Equality

Two sets $A$ and $B$ are equal if and only if each is a subset of the other: $A = B \iff A \subseteq B \text{ and } B \subseteq A$.

Cardinality

The number of elements in a finite set $A$, denoted by $|A|$, $\text{card}(A)$, or $\#A$.

Power Set

The set of all subsets of a set $A$, denoted $\mathcal{P}(A) := \{B \mid B \subseteq A\}$. If a finite set $A$ has $n$ elements, $\text{card}(\mathcal{P}(A)) = 2^n$.

Intersection

A set operation containing all elements that belong to both set $A$ and set $B$, defined as $A \cap B := \{x \mid x \in A \text{ and } x \in B\}$.

Union

A set operation containing all elements that belong to at least one of the sets $A$ or $B$, defined as $A \cup B := \{x \mid x \in A \text{ or } x \in B\}$.

Set Difference

The set of all elements of $A$ that are not elements of $B$, denoted $A \setminus B$ or $A - B$, defined as $A \setminus B := \{x \mid x \in A \text{ and } x \notin B\}$.

Symmetric Difference

The set containing all elements that belong to exactly one of two sets $A$ and $B$, defined as $A \Delta B := (A \setminus B) \cup (B \setminus A)$.

Complement

The set containing all elements of a universe $U$ that are not elements of $A$, defined as $A^c := \{x \in U \mid x \notin A\} = U \setminus A$.

De Morgan's Laws (Sets)

The identities relating unions and intersections via complementation: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

Cartesian Product

The set of all ordered pairs $(a, b)$ formed from two sets $A$ and $B$, defined as $A \times B := \{(a, b) \mid a \in A, b \in B\}$. In general, $A \times B \neq B \times A$.

Element Table

A verification method analogous to truth tables in propositional logic, used to prove set identities by systematically listing element membership ($\in$ or $\notin$) across all combinations.