Practice Symmetry in Operations in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium
For addition, show that if a+b=c, then b+a=c (symmetry). Use a=12,b=7.

Example 2

hard
Is the operation a◊b=ab symmetric? Justify briefly.

Example 3

medium
Is a⋆b=a2+b2 symmetric? Justify.

Example 4

hard
Is the 'less than or equal' relation ≤ symmetric? If 3≤5, is 5≤3?

Example 5

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

Example 6

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

Example 7

easy
Show that for any a,b, a+b=b+a using a=14,b=9.

Example 8

easy
Does 8−5 equal 5−8? What does this tell us about subtraction's symmetry?

Example 9

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

Example 10

easy
Is 20−7 the same as 7−20?

Example 11

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

Example 12

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

Example 13

easy
Is exponentiation symmetric? Compare 23 vs 32.

Example 14

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 15

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

Example 16

easy
Is the relation 'less than' (<) symmetric? If 3<5, is 5<3?

Example 17

easy
Is the relation 'equals' symmetric? If a=b, does b=a?

Example 18

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

Example 19

medium
Use symmetry of multiplication to compute 25×17×4 quickly.

Example 20

easy
Does a+b=b+a require a=b?
← Back to Symmetry in Operations Worked Examples Formula Explained