# Sets and Set-Builder Notation

Mathematics I · Mathematical Language and Reasoning · https://tryals.app/learn/mathematics-i/sets-and-set-builder-notation

## Sets and Elements

A **set** is a well-defined collection of distinct objects called **elements**. Nearly all modern mathematics is built on sets.

| Notation | Meaning |
| --- | --- |
| $x \in A$ | $x$ is an element of $A$ |
| $x \notin A$ | $x$ is not an element of $A$ |
| $\emptyset$ | The **empty set**, containing no elements |

A set is determined entirely by its elements. Order and repetition are irrelevant, so $\{1, 2, 2, 3\} = \{1, 2, 3\}$.

Sets are specified in two ways:

- **Roster notation** lists elements directly: $\{2, 4, 6, 8\}$.
- **Set-builder notation** uses a rule: $\{x \in \mathbb{Z} : \text{even and } 0 < x < 10\}$.

Standard number sets have reserved symbols: $\mathbb{N}$ (naturals), $\mathbb{Z}$ (integers), $\mathbb{Q}$ (rationals), and $\mathbb{R}$ (reals).

## Subsets and Power Sets

Key relationships define how sets interact with each other and their own sizes:

- **Subset** ($A \subseteq B$): every element of $A$ is also in $B$. The empty set $\emptyset$ is a subset of _every_ set.
- **Cardinality** ($|A|$): the number of elements in $A$.
- **Power set** ($\mathcal{P}(A)$): the set of all subsets of $A$. If $|A| = n$, then the cardinality is given by $|\mathcal{P}(A)| = 2^{|A|}$.

> **Pitfall:** Do not confuse **membership** ($\in$) with the **subset** relation ($\subseteq$).

$3 \in \{1,2,3\}$ is true because $3$ is an element. However, $\{3\} \subseteq \{1,2,3\}$ is true because $\{3\}$ is a set. Writing $3 \subseteq \{1,2,3\}$ is nonsense because $3$ is not a set.

## Practice questions

7 of this lesson's 11 practice questions, with answers. The full set is in the app.

### 1. For $A = \{1, 2\}$, both $1 \in A$ and $\{1\} \in A$ are true.

**Answer:** False

**Why:** False. The members of $A$ are 1 and 2, so $1 \in A$ holds but $\{1\} \in A$ does not, $\{1\}$ is a *subset*, written $\{1\} \subseteq A$. Membership and inclusion are different relations, and the distinction becomes essential once sets contain other sets.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/for-a-1-2-both-1-a-and-1-a-are-true

### 2. Is there a set of **all** sets? Russell asked instead about the set of all sets that do not contain themselves. Why is that question fatal?

A. Because it contains itself exactly if it does not
B. Because the description is too vague to pick out a definite collection
C. Because sets cannot contain other sets as elements
D. Because such a set would be infinite, and infinite sets are not permitted

**Answer:** A. Because it contains itself exactly if it does not

**Why:** Both answers refute themselves, and the description was flawless, so the fault is in the assumption that any description defines a set. Modern axioms build sets from existing ones instead, which is the whole reason set theory looks so bureaucratic: it is arranged to make this sentence unwriteable.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/is-there-a-set-of-all-sets-russell-asked-instead-about-the-set-of

### 3. For any set $A$, both $\emptyset \subseteq A$ and $\emptyset \in A$ hold.

**Answer:** False

**Why:** False. $\emptyset \subseteq A$ always holds, vacuously, since there is no element of $\emptyset$ to find missing from $A$. But $\emptyset \in A$ requires the empty set to be *listed* as a member, which is true only for sets that happen to contain it, such as $\{\emptyset\}$. Inclusion and membership are different relations.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/for-any-set-a-both-subseteq-a-and-a-hold

### 4. A set with 3 elements has 8 subsets because each element is independently either in or out.

**Answer:** True

**Why:** True. Each element contributes an independent in-or-out choice, giving $2^{n}$ subsets, 8 for three elements. The same counting argument shows why the power set is strictly larger than the set, which is the finite shadow of Cantor's theorem.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/a-set-with-3-elements-has-8-subsets-because-each-element-is

### 5. Sort each object by whether it belongs to the set S = { x : x is an integer and x squared < 10 }.

**Answer:**

- In S: 3, -3, 0
- Not in S: 4, 1.5, -4

**Why:** S = {-3, -2, -1, 0, 1, 2, 3}. Both 3 and -3 square to 9 < 10, so both qualify. 4 and -4 square to 16, too big. And 1.5 fails the very first condition, it is not an integer, even though its square is only 2.25.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/sort-each-object-by-whether-it-belongs-to-the-set-s-x-x-is-an

### 6. $\{x : x \in \mathbb{R},\ x^{2} = -1\}$ and $\{x : x \in \mathbb{Z},\ 0 < x < 1\}$ are both empty. Why is there only ONE empty set rather than two?

A. Because a set is determined entirely by its members
B. They are in fact different sets, distinguished by their defining conditions
C. Because both conditions are contradictory in the same way
D. Because the empty set is defined as a subset of every set

**Answer:** A. Because a set is determined entirely by its members

**Why:** Extensionality: same members, same set. The two conditions are different sentences picking out the same object, which is why "the" empty set is correct.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/x-x-x-1-and-x-x-0-x-1-are-both-empty-why

### 7. For A = {1, 2}, the statement "{1} is a subset of A" is true while "{1} is an element of A" is false. Why does putting braces around the 1 change which relation holds?

A. Because braces are optional notation, so the two statements say the same thing and one has simply been miswritten.
B. Because subset and element mean the same relation, applied to sets and to numbers respectively.
C. Because {1} is a subset of every set, whereas being an element depends on the particular set.
D. Because A's elements are 1 and 2, not sets, and {1} is a set.

**Answer:** D. Because A's elements are 1 and 2, not sets, and {1} is a set.

**Why:** A contains exactly two things, the numbers 1 and 2, and {1} is neither, but every member of {1} lies in A, which is what subset means. The distinction becomes unmistakable with A = {1, {1}}, where {1} is both, for different reasons. Set theory builds everything by nesting, so containing and being-among-your-members must stay apart.

Page: https://tryals.app/practice/mathematics-i/sets-and-set-builder-notation/for-a-1-2-the-statement-1-is-a-subset-of-a-is-true-while