For all real numbers x and y, the custom operation \odot is defined by x \odot y = x|y| - … — GRE General Test

For all real numbers $x$ and $y$ , the custom operation $\odot$ is defined by $x \odot y = x|y| - y|x|$ . Which of the following statements must be true for all real numbers $x$ and $y$ ? Select all such statements.

$x \odot y = 0$ whenever $x$ and $y$ have the same signAnswer
$x \odot y = -(y \odot x)$ Answer
If $x > 0$ and $y < 0$ , then $x \odot y > 0$ Answer
D
$x \odot (-x) = 0$ for all non-zero real numbers $x$
E
$(x \odot y) \odot z = x \odot (y \odot z)$

Answer

The correct statements are the statement asserting

x \odot y = 0

when

x

and

y

have the same sign, the statement asserting anti-commutativity

x \odot y = -(y \odot x)

, and the statement asserting

x \odot y > 0

when

x > 0

and

y < 0

The operation

x \odot y = x|y| - y|x|

produces 0 whenever

x

and

y

share the same sign because terms evaluate to identical quantities. Swapping variables negates the expression, establishing anti-commutativity. When

x

is positive and

y

is negative,

x \odot y

simplifies to

-2xy

, which is strictly greater than 0 since

xy < 0

Step-by-Step Solution

Analyze the first statement regarding same-sign inputs

x > 0

and

y > 0

|x|=x

and

|y|=y

, so

x \odot y = xy - yx = 0

. If

x < 0

and

y < 0

|x|=-x

and

|y|=-y

, so

x \odot y = x(-y) - y(-x) = -xy + xy = 0

. Thus,

x \odot y = 0

when

x

and

y

have the same sign.

Verifying the definition under both positive and negative cases of identical sign.

Analyze the second statement regarding operand order reversal

y \odot x = y|x| - x|y| = -(x|y| - y|x|) = -(x \odot y)

, which holds universally for all real numbers.

Testing anti-commutativity by algebraic substitution into the custom operation.

Analyze the third statement for opposite signs (

x > 0

and

y < 0

)

Since

x > 0

|x|=x

. Since

y < 0

|y|=-y

. Substituting yields

x(-y) - y(x) = -xy - xy = -2xy

. Because

x > 0

and

y < 0

, the product

xy

is negative, making

-2xy

strictly positive.

Determining the overall algebraic sign of the expression when variables have opposite signs.

Counter-test the remaining statements to verify incorrectness

For

x \odot (-x)

with

x=1

1 \odot (-1) = 1|-1| - (-1)|1| = 1 - (-1) = 2 \neq 0

. For associativity with

x=2, y=-1, z=-1

(2 \odot -1) \odot -1 = 4 \odot -1 = 8

, but

2 \odot (-1 \odot -1) = 2 \odot 0 = 0

Demonstrating specific counterexamples for false generalizations.

Key Concept

Custom Binary Operations and Absolute Value Properties

Estimated Time:1m 30s

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