Algebra as Structure Examples in Math

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

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

Concept Recap

The perspective that algebra is the systematic study of abstract mathematical structures and the operations defined on them.

Beyond numbers: what happens when ANY set has operations with certain properties?

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: Algebra-as-structure studies sets-with-operations by the properties they obey, not the specific elements.

Common stuck point: The procedure for algebra as structure is the easy part; the trap is assuming every operation is commutative or associative. Asking "Is the question about the properties of an operation, rather than computing a number?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the question about the properties of an operation, rather than computing a number?

Worked Examples

Example 1

medium

Show that addition of integers and multiplication of integers share a structural property (commutativity).

Answer

Both are commutative: order doesn't matter.

First step

Step 1: Addition:

a + b = b + a

for all integers. Example:

3 + 5 = 5 + 3

See the full worked solution + why-it-works coaching

SetupKey insightWhy it worksCommon pitfallConnection

Unlock answer keys One Family plan — every worked solution, all subjects

Example 2

hard

Does matrix multiplication have the same structural properties as number multiplication? Check commutativity.

Example 3

medium

Does the set

\{0,1,2,3\}

form a group under addition modulo 4? Verify the four group axioms.

Example 4

medium

Show that for

2 \times 2

matrices

A = \begin{pmatrix}0&1\\0&0\end{pmatrix}

and

B = \begin{pmatrix}0&0\\1&0\end{pmatrix}

AB \ne BA

Example 5

hard

Show that the symmetric difference

A \triangle B = (A \cup B) \setminus (A \cap B)

on the subsets of a fixed set

S

is associative.

Example 6

challenge

Define

a*b = a + b - ab

\mathbb{R} \setminus \{1\}

. Show it forms a group; find the identity and the inverse of

a

Practice Problems

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

Example 1

easy

What is the identity element for addition? For multiplication?

Example 2

medium

Does subtraction have an identity element?

Example 3

easy

Is addition of real numbers commutative? (Does

a+b=b+a

always hold?)

Example 4

easy

Is matrix multiplication commutative in general?

Example 5

easy

What is the identity element for multiplication of real numbers?

Example 6

easy

What is the additive inverse of 7 in the integers?

Example 7

easy

Is the set of integers closed under addition? (Is

a+b

always an integer?)

Example 8

easy

Are the integers closed under division? (Is

a/b

always an integer for integers

a,b\ne0

Example 9

easy

Which property does

a(b+c)=ab+ac

express?

Example 10

easy

Is subtraction of integers associative? Test with

a=8,b=3,c=2

: does

(a-b)-c=a-(b-c)

Example 11

medium

Do the even integers form a closed set under multiplication? Justify with the structural form.

Example 12

medium

\{1,-1\}

a group under multiplication? Check closure, identity, and inverses.

Example 13

medium

Find the multiplicative inverse of 3 modulo 7 (the structure

\mathbb{Z}_7

Example 14

medium

Is the operation

a*b=a+b-1

on the reals associative? Check

(a*b)*c

a*(b*c)

Example 15

medium

In the structure

(\mathbb{R},*)

with

a*b=a+b-1

, find the identity element.

Example 16

medium

Why does the same axiom 'associativity' appear in integers-under-addition, matrices-under-multiplication, and function composition?

Example 17

medium

(\mathbb{Z},-)

(integers under subtraction) a group? Identify which axiom fails.

Example 18

medium

Is the set

\{0,1,2\}

under addition modulo 3 closed? Check

2+2 \bmod 3

Example 19

medium

Does the operation 'maximum' on real numbers,

a*b=\max(a,b)

, have an identity element? If so, what is it (consider extended reals)?

Example 20

challenge

Prove that in any group, the identity element is unique.

Example 21

challenge

Determine whether

\mathbb{Z}_6

under multiplication forms a group, and explain using the inverse axiom.

Example 22

challenge

Show that if

a*b=a+b+ab

\{x: x\ne-1\}

, the element

0

is the identity and find the inverse of

a

Example 23

easy

Is multiplication of real numbers associative? Test

(2 \cdot 3) \cdot 4

versus

2 \cdot (3 \cdot 4)

Example 24

easy

Is the set of natural numbers

\mathbb{N}

closed under subtraction? Give a counterexample if not.

Example 25

easy

What is the multiplicative inverse of

5

\mathbb{Q}

Example 26

easy

Which axiom does

5 + 3 = 3 + 5

illustrate?

Example 27

easy

Compute

4 \bmod 3

and

7 \bmod 3

, and check that

\mathbb{Z}_3

contains

\{0,1,2\}

Example 28

medium

\mathbb{Z}_5

, find the multiplicative inverse of

2

Example 29

medium

Is the set of odd integers closed under addition? Justify.

Example 30

medium

In a group with operation

*

, simplify

(a*b)^{-1}

Example 31

medium

Does

(\mathbb{Q},+)

have an identity element? If so, what is it?

Example 32

medium

Does the set

\{1,2,3,4\}

form a group under multiplication modulo 5?

Example 33

medium

Why is

0

not invertible under multiplication in any field?

Example 34

medium

Verify the distributive law for the specific values

a=2, b=3, c=4

: compute

a(b+c)

and

ab+ac

Example 35

hard

Show that the cancellation law

ax = ay \implies x = y

holds in any group.

Example 36

hard

\mathbb{Z}_6

, identify all elements that are not units (have no multiplicative inverse).

Example 37

hard

(\mathbb{R} \setminus \{0\}, \times)

a group? Verify the axioms.

Example 38

hard

Is the set of

2 \times 2

invertible matrices over

\mathbb{R}

a group under multiplication? Name it.

Example 39

hard

Are the polynomials

\mathbb{Z}[x]

closed under addition and multiplication?

Example 40

challenge

Prove: in any group,

(a^{-1})^{-1} = a

Example 41

challenge

Show that

\mathbb{Z}_n

is a field if and only if

n

is prime.

← Back to Algebra as Structure Practice Mode

Related Concepts

Expressions

Background Knowledge

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

expressions