Symmetry in Operations Examples: 45 Problems with Answers

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≠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 5×3=3×5. What symmetric property do both share?

Answer

Both equal the same value; both are commutative

First step

1
5+3=8 and 3+5=8. Equal! ✓

Full solution

  1. 2
    5×3=15 and 3×5=15. Equal! ✓
  2. 3
    Both operations are symmetric (commutative): swapping inputs gives the same output.
  3. 4
    This is the commutative property for both addition and multiplication.
Operations with symmetry (commutativity) satisfy a∘b=b∘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 as (5+7)−(3+2). Is this rearrangement valid? Verify.

Example 7

challenge
Define a⊕b=(a+b)2. Prove ⊕ 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
Is 16÷4=4÷16? What does this tell us about division's symmetry?

Example 3

easy
Is addition symmetric? Check 3+5 vs 5+3.

Example 4

easy
Is subtraction symmetric? Check 3−5 vs 5−3.

Example 5

easy
Is multiplication symmetric? Check 4×6 vs 6×4.

Example 6

easy
Is division symmetric? Check 8÷2 vs 2÷8.

Example 7

easy
Which of +,−,×,÷ 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⋆b=a+b, is it symmetric? Justify generally.

Example 12

medium
Is f(x)=x+1 symmetric as a graph about the y-axis, even though addition is symmetric?

Example 13

medium
Is the operation a⋆b=∣a−b∣ symmetric? Test and justify.

Example 14

medium
Is exponentiation ab symmetric? Test with 2 and 4.

Example 15

medium
Is the operation 'average', a⋆b=a+b2, 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
Is a⋆b=a2+b2 symmetric? Justify.

Example 19

challenge
Determine all a,b for which division is 'symmetric', i.e. a÷b=b÷a (with a,b≠0).

Example 20

challenge
Show that a⋆b=ab−a−b is symmetric and find a value where a⋆b=b⋆a=0.

Example 21

challenge
A relation ∼ is symmetric and a∼b holds. If also a∼a always holds, what two relation properties are present, and is ∼ necessarily transitive?

Example 22

medium
Is the operation a⋆b=ab+1 symmetric? Justify.

Example 23

easy
Is 11+9=9+11?

Example 24

easy
Is 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 25×17×4 quickly.

Example 28

medium
When does a÷b=b÷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∘b=a−2b symmetric? Check with a=4,b=1.

Example 32

medium
Is the operation a∘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 ≤ symmetric? If 3≤5, is 5≤3?

Example 35

hard
Define a⋆b=a2+b2. Is ⋆ 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 4×7×25 mentally.

Example 38

challenge
For how many ordered pairs of digits (a,b) with a,b∈{1,2,3,4,5} is ab=ba?
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Background Knowledge

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

commutativity