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Difficulty: Very hardOrder of Operations and Number Properties

For all real numbers xx and yy, two custom operations, ⊕\oplus and ⊗\otimes, are defined as follows:

x⊕y=x+y−1x \oplus y = x + y - 1
x⊗y=xy−x−y+2x \otimes y = xy - x - y + 2

Which of the following statements about these operations must be true for all real numbers aa, bb, and cc?

I. a⊕(b⊕c)=(a⊕b)⊕ca \oplus (b \oplus c) = (a \oplus b) \oplus c
II. a⊗b=b⊗aa \otimes b = b \otimes a
III. a⊗(b⊕c)=(a⊗b)⊕(a⊗c)a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c)

  1. A
    I only
  2. B
    II only
  3. C
    I and II only
  4. D
    II and III only
  5. I, II, and IIIAnswer

Answer

The statement containing all three Roman numerals, I, II, and III, is correct because the associative, commutative, and distributive properties all hold true under these custom definitions.
All three statements are true. First, the operation ⊕\oplus is associative because both a⊕(b⊕c)a \oplus (b \oplus c) and (a⊕b)⊕c(a \oplus b) \oplus c simplify to a+b+c−2a + b + c - 2. Second, the operation ⊗\otimes is commutative because a⊗b=ab−a−b+2a \otimes b = ab - a - b + 2 and b⊗a=ba−b−a+2b \otimes a = ba - b - a + 2 are algebraically identical. Third, the operation ⊗\otimes distributes over ⊕\oplus because both a⊗(b⊕c)a \otimes (b \oplus c) and (a⊗b)⊕(a⊗c)(a \otimes b) \oplus (a \otimes c) simplify to ab+ac−2a−b−c+3ab + ac - 2a - b - c + 3. Therefore, the statement including all three Roman numerals is correct.

Step-by-Step Solution

1
Evaluate Statement I for associativity by expanding a⊕(b⊕c)a \oplus (b \oplus c) and (a⊕b)⊕c(a \oplus b) \oplus c.
Both expressions simplify to a+b+c−2a + b + c - 2.
Since both groupings yield the same algebraic expression, the operation ⊕\oplus is associative.
2
Evaluate Statement II for commutativity by comparing a⊗ba \otimes b and b⊗ab \otimes a.
a⊗b=ab−a−b+2a \otimes b = ab - a - b + 2 and b⊗a=ba−b−a+2b \otimes a = ba - b - a + 2.
Since multiplication and addition of real numbers are commutative (ab=baab = ba and −a−b=−b−a-a - b = -b - a), the two expressions are equal, meaning the operation ⊗\otimes is commutative.
3
Evaluate Statement III for distributivity by expanding both sides of a⊗(b⊕c)=(a⊗b)⊕(a⊗c)a \otimes (b \oplus c) = (a \otimes b) \oplus (a \otimes c).
The left side expands to ab+ac−2a−b−c+3ab + ac - 2a - b - c + 3. The right side also expands to ab+ac−2a−b−c+3ab + ac - 2a - b - c + 3.
Because both sides simplify to the exact same expression, the operation ⊗\otimes distributes over ⊕\oplus.

Key Concept

Testing the definitions of the commutative, associative, and distributive properties using custom operations.
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Topics

Order of Operations and Number Properties

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