Chapter 1: Sets, Logic and Functions - Vocabulary

Vocabulary practice flashcards covering fundamental set concepts, notations, set types, formulas, set operations, and set laws from Chapter 1.

Set

A collection of unordered objects called elements.

Element Symbol ($\in$)

A notation used to indicate that an object $a$ is an element of set $A$ (written $a \in A$).

Non-Element Symbol ($\notin$)

A notation used to indicate that an object $a$ is not an element of set $A$ (written $a \notin A$).

Empty Set

A set that contains no elements, denoted as $\emptyset$ or $\{\}$.

Roster Method

A method of describing a set by explicitly listing its elements inside curly brackets, such as $S = \{a, b, c, d\}$.

Set-Builder Notation

A notation that uses a condition to describe the elements of a set, formatted as $S = \{x \mid \text{condition on } x\}$.

Natural Numbers ($\mathbb{N}$)

The set of non-negative counting numbers, represented as $\mathbb{N} = \{0, 1, 2, 3, \dots\}$.

Integers ($\mathbb{Z}$)

The set of whole numbers containing negative integers, zero, and positive integers, represented as $\mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\}$.

Positive Integers ($\mathbb{Z}^+$)

The set of all integers strictly greater than zero, represented as $\mathbb{Z}^+ = \{1, 2, 3, 4, \dots\}$.

Rational Numbers ($\mathbb{Q}$)

The set of numbers that can be written as a fraction of two integers, represented as $\mathbb{Q} = \{1.5, 2.6, -3.8, 15, \dots\}$.

Real Numbers ($\mathbb{R}$)

The set containing positive numbers, negative numbers, zero, fractions, and irrational numbers, represented as $\mathbb{R} = \{47.3, -2.5, \pi, \dots\}$.

Subset ($A \subseteq B$)

A set $A$ where every element of $A$ is also in $B$, formally defined as $A \subseteq B \iff \forall x (x \in A \rightarrow x \in B)$.

Proper Subset ($A \subset B$)

A set $A$ that is a subset of $B$ ($A \subseteq B$) where there exists at least one element in $B$ that is not in $A$.

Number of Subsets Formula

The total number of distinct subsets of a set containing $n$ elements, given by $2^n$.

Number of Proper Subsets Formula

The total number of distinct proper subsets of a set containing $n$ elements, given by $2^n - 1$.

Cardinality ($|A|$)

The number of distinct elements in a finite set $A$.

Power Set ($P(S)$)

The collection of all subsets of a set $S$, including the empty set and the original set itself, with cardinality $|P(S)| = 2^n$.

Equal Sets

Two sets $A$ and $B$ that have the exact same elements, formally written as $A = B \iff (A \subseteq B) \land (B \subseteq A)$.

Union ($A \cup B$)

The set containing all elements from both sets $A$ and $B$, defined as $A \cup B = \{x \mid x \in A \lor x \in B\}$.

Intersection ($A \cap B$)

The set containing the common elements that appear in both set $A$ and set $B$, defined as $A \cap B = \{x \mid x \in A \land x \in B\}$.

Set Difference ($A - B$)

The set containing exactly those elements of $A$ that are not in $B$, defined as $A - B = \{x \mid x \in A \land x \notin B\}$.

Complement ($A^c$)

The set containing all elements in the universal set $U$ that do not belong to $A$, calculated as $A^c = U - A$.

Disjoint Sets

Two sets $A$ and $B$ that have no elements in common, meaning $A \cap B = \emptyset$.

Commutative Laws for Sets

Laws stating that order does not change set operations: $A \cup B = B \cup A$ and $A \cap B = B \cap A$.

Associative Laws for Sets

Laws stating that grouping does not change set operations: $A \cup (B \cup C) = (A \cup B) \cup C$ and $A \cap (B \cap C) = (A \cap B) \cap C$.

Distributive Laws for Sets

Laws stating that union and intersection distribute over each other: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

De Morgan's Laws for Sets

Laws relating complements of unions and intersections: $(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$.

Identity Laws for Sets

Laws stating that $A \cup \emptyset = A$ and $A \cap U = A$.

Domination Laws for Sets

Laws stating that $A \cup U = U$ and $A \cap \emptyset = \emptyset$.

Idempotent Laws for Sets

Laws stating that operating on a set with itself yields the set itself: $A \cup A = A$ and $A \cap A = A$.

Double Complement Law

Law stating that taking the complement of a complement returns the original set: $(A^c)^c = A$.