MAT 1310: Introduction to the Theory of Sets

Flashcards covering the fundamentals of set theory, including definitions, set operations, algebraic laws, number classifications (rational, irrational, prime), and interval notation based on Mr. Anthony Katakwe's lecture notes.

According to Definition 1, what is a set?

A set is a collection of distinct elements or objects.

If an element $x$ is in a set $A$, what is the notation used and how is it read?

It is written as $x \in A$ and read as "$x$ is a member (element) of $A$" or "$x$ belongs to $A$".

What are the two methods for describing a set mentioned in the notes?

(i) Roster/Listing Method/Tabular Form (listing elements explicitly) and (ii) Rule Method/Set-builder (specifying a property).

How is a set classified if it contains a finite number of elements or no elements?

It is said to be a finite set.

What is the definition of the cardinal number of a finite set, and how is it denoted?

The number of elements in a given finite set is called the cardinal number, denoted by $n(A)$.

What is an empty set (also called a null or void set)?

A set which contains no elements, denoted by $\emptyset$ or ${}$. It is considered finite and a subset of every other set.

What is a singleton set?

A singleton set is a set containing only a single element.

What condition must be met for set $A$ to be a subset of set $B$ ($A \subseteq B$)?

Every element of set $A$ must also belong to set $B$, meaning $x \in A$ implies $x \in B$.

Distinguish between equal sets and equivalent sets.

Two sets are equal ($A = B$) if they contain exactly the same elements. They are equivalent ($A \equiv B$) if they contain the same number of elements, meaning $n(A) = n(B)$.

What is the definition of a proper subset?

Set $A$ is a proper subset of $B$ if $A \subseteq B$ but there is at least one element of $B$ that is not in $A$ ($A \neq B$), written as $A \subset B$.

When are two sets $A$ and $B$ called disjoint?

They are disjoint if they do not have any common element, meaning $A \cap B = \emptyset$.

What is a universal set?

A set consisting of all possible elements which occur under consideration, denoted by $U$.

What is the formula for the number of subsets of a finite set with $n$ elements?

$$2^n$$

How is the union of sets $A$ and $B$ ($A \cup B$) defined?

The set of all elements which belong to either $A$ or $B$, written as $A \cup B = \{x : x \in A \text{ or } x \in B\}$.

How is the intersection of sets $A$ and $B$ ($A \cap B$) defined?

The set of all elements which belong to both $A$ and $B$, written as $A \cap B = \{x : x \in A \text{ and } x \in B\}$.

What is the complement of a set $A$ ($A^c$)?

The set of elements in the universal set $U$ that do not belong to $A$, written as $A^c = \{x : x \in U, x \notin A\}$.

What is the relative complement (or difference) of set $B$ with respect to set $A$ ($A \setminus B$)?

The set of elements which belong to $A$ but do not belong to $B$, written as $A \setminus B = \{x : x \in A \text{ and } x \notin B\} = A \cap B^c$.

What are De-Morgan’s Laws in set algebra?

$(A \cup B)^c = A^c \cap B^c$ and $(A \cap B)^c = A^c \cup B^c$

How is the Cartesian product of sets $A$ and $B$ ($A \times B$) defined?

$$A \times B = \{(a, b) : a \in A \text{ and } b \in B\}$$

What are rational numbers according to Definition 29?

Numbers in decimal form whose block of digits terminates or repeats, or numbers that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.

What is the definition of an irrational number?

A number in decimal form whose block of digits neither terminates nor repeats.

What is the difference between a prime number and a composite number?

A prime number is a positive integer greater than $1$ with no positive integer factors other than itself and $1$. A composite number is a positive integer greater than $1$ that is not prime.

In interval notation, what is represented by the interval $(a, b)$ and the interval $[a, b]$?

$(a, b)$ is an open interval representing all real numbers between $a$ and $b$ not including $a$ and $b$. $[a, b]$ is a closed interval representing all real numbers between $a$ and $b$ including $a$ and $b$.

What does the notation $[b, \infty)$ represent?

The set of all real numbers greater than or equal to $b$, or in set-builder notation: $\{x : x \ge b\}$.