Practice Symmetry in Operations in Math

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

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Quick Recap

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

3+5=5+33 + 5 = 5 + 3 shows addition is symmetric. 3โˆ’5โ‰ 5โˆ’33 - 5 \neq 5 - 3 shows subtraction isn't.

Showing a random 20 of 50 problems.

Example 1

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

Example 2

hard
Is the operation aโ—Šb=aba \Diamond b = a^b symmetric? Justify briefly.

Example 3

medium
Is aโ‹†b=a2+b2a\star b = a^2+b^2 symmetric? Justify.

Example 4

hard
Is the 'less than or equal' relation โ‰ค\le symmetric? If 3โ‰ค53 \le 5, is 5โ‰ค35 \le 3?

Example 5

medium
Is the operation aโ‹†b=ab+1a\star 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,ba, b, a+b=b+aa + b = b + a using a=14,b=9a = 14, b = 9.

Example 8

easy
Does 8โˆ’58 - 5 equal 5โˆ’85 - 8? What does this tell us about subtraction's symmetry?

Example 9

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

Example 10

easy
Is 20โˆ’720 - 7 the same as 7โˆ’207 - 20?

Example 11

challenge
For how many ordered pairs of digits (a,b)(a, b) with a,bโˆˆ{1,2,3,4,5}a, b \in \{1,2,3,4,5\} is ab=baa^b = b^a?

Example 12

challenge
Show that aโ‹†b=abโˆ’aโˆ’ba\star b=ab-a-b is symmetric and find a value where aโ‹†b=bโ‹†a=0a\star b=b\star a=0.

Example 13

easy
Is exponentiation symmetric? Compare 232^3 vs 323^2.

Example 14

challenge
A relation โˆผ\sim is symmetric and aโˆผba\sim b holds. If also aโˆผaa\sim a always holds, what two relation properties are present, and is โˆผ\sim necessarily transitive?

Example 15

medium
Is the operation aโ‹†b=โˆฃaโˆ’bโˆฃa\star b=|a-b| symmetric? Test and justify.

Example 16

easy
Is the relation 'less than' (<<) symmetric? If 3<53<5, is 5<35<3?

Example 17

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

Example 18

medium
Is the operation 'average', aโ‹†b=a+b2a\star b=\frac{a+b}{2}, symmetric?

Example 19

medium
Use symmetry of multiplication to compute 25ร—17ร—425 \times 17 \times 4 quickly.

Example 20

easy
Does a+b=b+aa+b=b+a require a=ba=b?
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