Symmetry in Operations Examples in Math

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

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

Concept Recap

When exchanging or swapping operands or roles in an operation produces the same result or a symmetrically related one.

$3 + 5 = 5 + 3$ shows addition is symmetric. $3 - 5 \neq 5 - 3$ shows subtraction isn't.

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 operation is symmetric when exchanging the two inputs leaves the result unchanged, like $a+b=b+a$ .

Common stuck point: The procedure for symmetry in operations is the easy part; the trap is assuming every operation is symmetric. Asking "Does exchanging the two inputs leave the result exactly the same?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does exchanging the two inputs leave the result exactly the same?

Worked Examples

Example 1

easy

Show that

5 + 3 = 3 + 5

and

5 \times 3 = 3 \times 5

. What symmetric property do both share?

Answer

Both equal the same value; both are commutative

First step

5 + 3 = 8

and

3 + 5 = 8

. Equal! ✓

Full solution

2
$5 \times 3 = 15$ and $3 \times 5 = 15$ . Equal! ✓
3
Both operations are symmetric (commutative): swapping inputs gives the same output.
4
This is the commutative property for both addition and multiplication.

Operations with symmetry (commutativity) satisfy

a \circ b = b \circ a

. Addition and multiplication both have this symmetry.

Example 2

medium

For addition, show that if

a + b = c

, then

b + a = c

(symmetry). Use

a=12, b=7

Example 3

easy

Show that for any

a, b

a + b = b + a

using

a = 14, b = 9

Example 4

medium

Show that subtraction can sometimes give equal results when swapped: when does

a - b = b - a

Example 5

hard

Show by example that

(a+b) + c = a + (b+c)

using

a=3, b=7, c=5

. Is this symmetry or associativity?

Example 6

hard

A student computes

5 - 3 + 7 - 2

(5+7) - (3+2)

. Is this rearrangement valid? Verify.

Example 7

challenge

Define

a \oplus b = (a+b)^2

. Prove

\oplus

is symmetric.

Practice Problems

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

Example 1

easy

Does

8 - 5

equal

5 - 8

? What does this tell us about subtraction's symmetry?

Example 2

medium

16 \div 4 = 4 \div 16

? What does this tell us about division's symmetry?

Example 3

easy

Is addition symmetric? Check

3+5

5+3

Example 4

easy

Is subtraction symmetric? Check

3-5

5-3

Example 5

easy

Is multiplication symmetric? Check

4\times 6

6\times 4

Example 6

easy

Is division symmetric? Check

8\div 2

2\div 8

Example 7

easy

Which of

+,-,\times,\div

are symmetric?

Example 8

easy

Is the relation 'equals' symmetric? If

a=b

, does

b=a

Example 9

easy

Is the relation 'less than' (

<

) symmetric? If

3<5

, is

5<3

Example 10

easy

Does

a+b=b+a

require

a=b

Example 11

medium

For the operation

a\star b=a+b

, is it symmetric? Justify generally.

Example 12

medium

f(x)=x+1

symmetric as a graph about the

y

-axis, even though addition is symmetric?

Example 13

medium

Is the operation

a\star b=|a-b|

symmetric? Test and justify.

Example 14

medium

Is exponentiation

a^b

symmetric? Test with 2 and 4.

Example 15

medium

Is the operation 'average',

a\star b=\frac{a+b}{2}

, symmetric?

Example 16

medium

Is the relation 'is a sibling of' symmetric? Is 'is the parent of' symmetric?

Example 17

medium

Which property of the operation makes a multiplication table symmetric across its diagonal?

Example 18

medium

a\star b = a^2+b^2

symmetric? Justify.

Example 19

challenge

Determine all

a,b

for which division is 'symmetric', i.e.

a\div b=b\div a

(with

a,b\ne 0

Example 20

challenge

Show that

a\star b=ab-a-b

is symmetric and find a value where

a\star b=b\star a=0

Example 21

challenge

A relation

\sim

is symmetric and

a\sim b

holds. If also

a\sim a

always holds, what two relation properties are present, and is

\sim

necessarily transitive?

Example 22

medium

Is the operation

a\star b = ab+1

symmetric? Justify.

Example 23

easy

11 + 9 = 9 + 11

Example 24

easy

20 - 7

the same as

7 - 20

Example 25

easy

Is the relation 'is the parent of' symmetric?

Example 26

medium

Use symmetry of addition to compute

98 + 47 + 2

quickly.

Example 27

medium

Use symmetry of multiplication to compute

25 \times 17 \times 4

quickly.

Example 28

medium

When does

a \div b = b \div a

for nonzero

a, b

Example 29

medium

Is the operation 'average of two numbers' symmetric? Check with

a = 6, b = 10

Example 30

medium

Is 'maximum of two numbers' symmetric?

Example 31

medium

Is the operation

a \circ b = a - 2b

symmetric? Check with

a=4, b=1

Example 32

medium

Is the operation

a \circ b = a + b - ab

symmetric? Check with

a=2, b=3

Example 33

hard

Given

a + b = b + a

always holds, what can you conclude about the order of terms in

5 + (3 + 8)

Example 34

hard

Is the 'less than or equal' relation

\le

symmetric? If

3 \le 5

, is

5 \le 3

Example 35

hard

Define

a \star b = a^2 + b^2

. Is

\star

symmetric?

Example 36

hard

A vending machine charges \$1 then dispenses a snack. Is the order of 'pay' and 'dispense' symmetric? What's the everyday lesson?

Example 37

hard

Use symmetry of multiplication to compute

4 \times 7 \times 25

mentally.

Example 38

challenge

For how many ordered pairs of digits

(a, b)

with

a, b \in \{1,2,3,4,5\}

a^b = b^a

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Related Concepts

Commutativity Even and Odd Functions Algebraic Symmetry

Background Knowledge

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

commutativity