Set Theory - Algebra & Trigonometry Module

Vocabulary practice flashcards covering basic set theory terminology, notations, set operations, properties, and laws from Unit 3 of the Algebra & Trigonometry Module at Malawi University of Science and Technology.

Set

A collection of objects or things which have at least one common attribute.

Elements (or Members)

The individual objects or things contained within a set.

Set-builder notation

A mathematical notation for expressing a set by describing its properties, written in forms such as $\{x : x \text{ is a prime number}\}$ or $\{x \mid x \text{ is a prime number}\}$.

Subset

A set $A$ is a subset of set $B$ (written $A \subseteq B$ or $B \supseteq A$) if every element of $A$ is also an element of $B$.

Equal sets

Two sets $A$ and $B$ that contain identical elements, defined formally as $A = B$ if $A \subseteq B$ and $B \subseteq A$.

Proper subset

A set $X$ that is entirely contained within a set $Y$ without being equal to $Y$, written as $X \subset Y$.

Cardinal number

The number of distinct (different) elements in a set $A$, denoted by $n(A)$ or $|A|$.

Union of Sets

The set of all elements that are in set $A$, in set $B$, or in both, denoted by $A \cup B = \{x : x \in A \text{ or } x \in B\}$.

Intersection of Sets

The set of all elements that are common to both set $A$ and set $B$, denoted by $A \cap B = \{x : x \in A \text{ and } x \in B\}$.

Universal Set

A single set containing all possible elements that may be referred to in a problem involving two or more sets, denoted by the symbol $\xi$ and represented visually by a rectangle in a Venn diagram.

Empty Set (Null Set)

A set that contains no elements, denoted by $\emptyset$ or $\{\}$, which is a subset of every set.

Disjoint sets

Two sets $A$ and $B$ that have no elements in common, meaning their intersection is the empty set ($A \cap B = \emptyset$).

Complement of a Set

Given a universal set $\xi$ and a set $A \subset \xi$, the complement $A'$ is the set of all elements in $\xi$ that do not belong to $A$.

De Morgan's Laws

Set theory laws stating that $(A \cap B)' = A' \cup B'$ and $(A \cup B)' = A' \cap B'$.

Difference Set

For any sets $P$ and $Q$, the difference set is $P - Q = \{x : x \in P \text{ and } x \notin Q\}$, consisting of elements in $P$ that are not in $Q$.

Symmetric Difference

For any sets $P$ and $Q$, the symmetric difference is $P \Delta Q = (P - Q) \cup (Q - P)$, representing elements belonging to either set but not to both.

Commutative Property of Sets

The property stating that the order of sets does not alter the result of union or intersection operations: $A \cup B = B \cup A$ and $A \cap B = B \cap A$.

Associative Property of Sets

The property stating that the grouping of sets does not alter the result of union or intersection operations: $(A \cup B) \cup C = A \cup (B \cup C)$ and $(A \cap B) \cap C = A \cap (B \cap C)$.