Element Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Element.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

An individual object that belongs to, or is a member of, a given set — either it is in the set or it is not.

An element is simply one item inside the collection — either it is in, or it is out. There is no "partially in."

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: An element is one individual member of a set, and membership is a sharp yes-or-no.

Common stuck point: The procedure for element is the easy part; the trap is mixing up $\in$ and $\subseteq$ . Asking "Am I asking about one single object being inside a set, with only a yes or no answer?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asking about one single object being inside a set, with only a yes or no answer?

Worked Examples

Example 1

easy

Let

A = \{3, 7, 11, 15\}

. Determine whether

7 \in A

\{7\} \in A

, and

10 \notin A

Answer

7 \in A,\quad \{7\} \notin A,\quad 10 \notin A

First step

Check

7 \in A

: the element

7

appears in the listing

\{3, 7, 11, 15\}

, so

7 \in A

. True.

Full solution

2
Check $\{7\} \in A$ : the object $\{7\}$ is a set, not a number. The set $A$ does not contain $\{7\}$ as a member, only the number $7$ . So $\{7\} \notin A$ .
3
Check $10 \notin A$ : $10$ does not appear in the listing, so indeed $10 \notin A$ . True.

The symbol

\in

tests whether an object is a direct member of a set. A set

\{7\}

and the number

7

are different objects — confusing them is the most common mistake with element notation.

Example 2

medium

Let

B = \{\emptyset, \{1\}, \{2, 3\}\}

. Which of the following are true? (a)

\emptyset \in B

, (b)

1 \in B

, (c)

\{1\} \in B

, (d)

\{2,3\} \subseteq B

Example 3

easy

Sort into elements and compounds:

\text{Fe}

\text{H}_2

\text{CO}_2

\text{NH}_3

\text{Ne}

Example 4

medium

A 50.0 g sample contains 25.0 g of element A and 25.0 g of element B, chemically bonded. Is it an element or compound?

Example 5

hard

A scientist isolates a substance that decomposes into 4.00 g of element A and 16.00 g of element B when heated, and confirms it has a fixed composition. Was the original substance an element?

Example 6

hard

Use mass conservation:

24\,\text{g}

of magnesium combines completely with oxygen to form

40\,\text{g}

of magnesium oxide. How many grams of oxygen reacted, and is MgO an element?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Let

S = \{1, 2, 3, 4, 5\}

. Fill in each blank with

\in

\notin

: (a)

3 \;\square\; S

, (b)

6 \;\square\; S

, (c)

0 \;\square\; S

Example 2

medium

Let

T = \{x \in \mathbb{Z} : x^2 < 10\}

. List all elements of

T

and state

|T|

Example 3

easy

2

an element of

\{1,2,3\}

? Write the symbol.

Example 4

easy

5 \in \{1,2,3\}

? Answer with the symbol.

Example 5

easy

Which symbol is correct:

2 \,?\, \{1,2,3\}

versus

\{2\} \,?\, \{1,2,3\}

Example 6

easy

1

the same as

\{1\}

Example 7

easy

a

an element of

\{a, b, c\}

Example 8

easy

How many elements are in

\{7\}

, and is

7

one of them?

Example 9

easy

0

an element of the set of natural numbers

\{1,2,3,\dots\}

as written here?

Example 10

easy

True or false: in

\{1,1,2\}

, the number 1 is still just one element.

Example 11

medium

For

A=\{1,\{2\},3\}

, decide: is

2 \in A

? Is

\{2\}\in A

Example 12

medium

For

B=\{\{1,2\},3\}

, is

1 \in B

? Explain.

Example 13

medium

\{1,2\}\subseteq\{1,2,3\}

? Is

\{1,2\}\in\{1,2,3\}

Example 14

medium

x \in A

and

A \subseteq B

, must

x \in B

Example 15

medium

List the elements of

A=\{n : n \text{ is an integer}, 1\le n\le 4\}

and confirm

3\in A

Example 16

medium

\emptyset \in \{\emptyset, 1\}

? Is

\emptyset \subseteq \{\emptyset, 1\}

Example 17

medium

For

A=\{1,2,3\}

, how many elements does the power set

\mathcal{P}(A)

have, and is

\{1,2\}\in\mathcal{P}(A)

Example 18

medium

\{a\} \in \{\{a\},\{b\}\}

? Is

a \in \{\{a\},\{b\}\}

Example 19

medium

Given

A=\{1,2\}

and

B=\{2,3\}

, list elements of

A\cup B

and confirm

2\in A\cap B

Example 20

challenge

For

A=\{\emptyset,\{\emptyset\}\}

, determine the truth of:

\emptyset\in A

\{\emptyset\}\in A

\{\emptyset\}\subseteq A

Example 21

challenge

Can a set be an element of itself in standard (ZFC) set theory? Explain via

A=\{x : x\notin x\}

Example 22

challenge

|A|=n

, how many subsets of

A

contain a fixed element

x

? Justify.

Example 23

easy

Which of these is a pure element:

\text{H}_2\text{O}

\text{Cu}

\text{NaCl}

\text{CH}_4

Example 24

easy

About how many elements are currently known?

Example 25

easy

Name an element that is a liquid at room temperature.

Example 26

easy

Is sulfur in

\text{S}_8

form considered an element?

Example 27

easy

Two atoms have 6 and 7 protons. Are they the same element?

Example 28

medium

Diamond and graphite both contain only carbon. Are they the same element? Are they the same substance?

Example 29

medium

Air contains nitrogen, oxygen, argon, and traces of others. Is air an element, compound, or mixture?

Example 30

medium

Stainless steel is made of iron, chromium, nickel, and carbon blended together. Is it an element?

Example 31

medium

Element X has atomic number 17 in one sample and atomic number 17 in another, but the two samples have masses 35 u and 37 u. Are they the same element?

Example 32

medium

Sodium (Na) reacts vigorously with chlorine (

\text{Cl}_2

) to form NaCl. Was a new element created?

Example 33

medium

An unknown shiny solid conducts electricity, can be hammered into sheets, and cannot be broken down chemically. What is it likely to be?

Example 34

hard

Why is hydrogen (H) sometimes placed in group 1 and sometimes in group 17 on the periodic table?

Example 35

hard

Element X has

Z = 79

. List its proton count, its likely conductivity, and one common use.

Example 36

hard

Element Q exists as a colorless gas that does not react with anything under normal conditions. To which family does it likely belong?

Example 37

hard

Compare an element (e.g.,

\text{O}_2

) and a compound (e.g.,

\text{O}_3

+ ozone) — both contain only oxygen. Why is

\text{O}_3

still considered an element?

Example 38

challenge

Helium (

\text{He}

) is the second-most abundant element in the universe but rare in Earth's atmosphere. Suggest one reason.

Example 39

challenge

Why is the periodic table organized by atomic number (

Z

) rather than by atomic mass?

← Back to Element Formula Explained Practice Mode

Related Concepts

Set Subset

Background Knowledge

These ideas may be useful before you work through the harder examples.

set