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

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

$a \odot b = b \odot a$ Answer
$a \odot 1 = 1$ Answer
C
$a \odot 0 = 0$
$(a \odot b) \odot c = a \odot (b \odot c)$ Answer
E
$a \odot a = a^2$

Answer

The statements

a \odot b = b \odot a

a \odot 1 = 1

, and

(a \odot b) \odot c = a \odot (b \odot c)

must be true for all real numbers

a

b

, and

c

The custom operation is commutative (

a \odot b = b \odot a

), maps any real number operated with

1

1

(

a \odot 1 = 1

), and is associative (

(a \odot b) \odot c = a \odot (b \odot c)

). These three properties hold universally for all real numbers.

Step-by-Step Solution

Evaluate

a \odot b

and

b \odot a

to test commutativity.

a \odot b = a + b - ab

and

b \odot a = b + a - ba

. Since addition and multiplication of real numbers are commutative,

a + b - ab = b + a - ba

Verify if order of operands alters the result.

Evaluate

a \odot 1

using the operation definition.

a \odot 1 = a + 1 - a(1) = a + 1 - a = 1

Test the behavior of operating with the constant

1

Evaluate

a \odot 0

to verify the zero property statement.

a \odot 0 = a + 0 - a(0) = a

. This equals

a

, not

0

for general values of

a

Check if operating with

0

results in

0

Evaluate both sides of

(a \odot b) \odot c = a \odot (b \odot c)

to test associativity.

Left side:

(a \odot b) \odot c = (a + b - ab) \odot c = (a + b - ab) + c - (a + b - ab)c = a + b + c - ab - ac - bc + abc

. Right side:

a \odot (b \odot c) = a \odot (b + c - bc) = a + (b + c - bc) - a(b + c - bc) = a + b + c - bc - ab - ac + abc

. Both sides are identical.

Verify whether grouping alters the result.

Evaluate

a \odot a

a \odot a = a + a - a(a) = 2a - a^2

. This is not equal to

a^2

except when

2a - a^2 = a^2 \implies 2a^2 - 2a = 0 \implies a=0

a=1

Determine if the self-operation yields

a^2

for all real numbers.

Key Concept

Evaluating algebraic properties (commutativity, identity, associativity) of custom binary operations.

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