The Numbers We Use: Sets and Subsets

Flashcards covering key definitions, mathematical set notations, subsets, and properties from the lesson on Sets and Subsets.

What is a set?

A set is a collection of objects, such as numbers, in which order does not matter.

What is a subset?

A subset is a set made up of objects from another set, called a parent set.

How many elements are in the set $\{-9, -8, -7\}$?

There are $3$ elements in the set.

How many elements are in the empty set, represented by $\{\}$ or $\emptyset$?

There are $0$ elements in the empty set.

Why is the set $\{7, 9, 12\}$ considered a subset of the set $\{5, 6, 7, 8, 9, 10, 11, 12\}$?

All elements of the set $\{7, 9, 12\}$ are contained within the set $\{5, 6, 7, 8, 9, 10, 11, 12\}$.

Why is the set $\{0, 1, 4, 9, 16\}$ not a subset of the set $\{1, 2, 3, 4, 5, \dots\}$?

Not all elements of the set $\{0, 1, 4, 9, 16\}$ are in $\{1, 2, 3, 4, 5, \dots\}$; specifically, $0$ is in the first set but not in the second.

What standard symbol represents the set of Natural Numbers?

$$\mathbb{N}$$

What standard symbol represents the set of Whole Numbers?

$$\mathbb{W}$$

What standard symbol represents the set of Integers?

$$\mathbb{Z}$$

What standard symbol represents the set of Rational Numbers?

$$\mathbb{Q}$$

What standard symbols are used to represent the set of Irrational Numbers?

$\mathbb{P}$, $\mathbb{Q}'$, or $\mathbb{R} \setminus \mathbb{Q}$

What standard symbol represents the set of Real Numbers?

$$\mathbb{R}$$

How are the Natural Numbers ($\mathbb{N}$) and Whole Numbers ($\mathbb{W}$) related?

Adding the element $0$ to the set of natural numbers gives the set of whole numbers. Therefore, $\mathbb{N}$ is a subset of $\mathbb{W}$.

How are the Whole Numbers ($\mathbb{W}$) and Integers ($\mathbb{Z}$) related?

The integers consist of the set of whole numbers along with the negative natural numbers. Therefore, $\mathbb{W}$ is a subset of $\mathbb{Z}$.

How are the Integers ($\mathbb{Z}$) and Rational Numbers ($\mathbb{Q}$) related?

The rational numbers consist of all numbers of the form $\frac{a}{b}$, where $a$ and $b$ are integers (with $b \neq 0$). Therefore, $\mathbb{Z}$ is a subset of $\mathbb{Q}$.

What is the relationship between Rational Numbers ($\mathbb{Q}$) and Irrational Numbers ($\mathbb{P}$)?

Rational numbers can be expressed as $\frac{a}{b}$ where $a$ and $b$ are integers, whereas irrational numbers cannot. They share no elements in common and are disjoint sets.

What does the notation $\mathbb{R} \setminus \mathbb{Q}$ signify?

It signifies the set of real numbers ($\mathbb{R}$) with the rational numbers ($\mathbb{Q}$) removed, leaving the set of irrational numbers.

What are all of the subsets of the set $\{1, 2\}$?

$\{\}$, $\{1\}$, $\{2\}$, and $\{1, 2\}$

Is a subset required to have strictly fewer elements than its parent set?

No. A subset can have the same number of elements as its parent set. (Only a proper subset must have strictly fewer elements.)

Are the sets $\{3, 4, 5\}$ and $\{4, 5, 3\}$ equal?

Yes. Two sets are equal if and only if they contain the exact same elements, regardless of the order in which they are written.

What formula gives the total number of possible subsets for a set containing $n$ elements?

$$2^n$$

How many possible subsets exist for a set containing $12$ elements?

$2^{12} = 4096$ subsets.

If set $C$ is a subset of set $B$ ($C \subseteq B$) and set $B$ is a subset of set $A$ ($B \subseteq A$), is set $C$ a subset of set $A$?

Yes. Because all elements of $C$ are contained in $B$, and all elements of $B$ are contained in $A$, all elements of $C$ are also contained in $A$ ($C \subseteq A$).

Is the empty set ($\emptyset$) a subset of every set?

Yes. A set containing no elements is a possible subset of any set.

Given $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $B = \{2, 4, 6, 8\}$, and $C = \{4, 8\}$, is the statement $B \subseteq A$ true or false?

True, because every element in $B$ ($2, 4, 6, 8$) is also in $A$.

Given $A = \{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, $B = \{2, 4, 6, 8\}$, and $C = \{4, 8\}$, is the statement $C \subseteq B$ true or false?

True, because every element in $C$ ($4, 8$) is also in $B$.