Let's analyze the problem step-by-step:
1. Define the operation:
The operation [tex]\( \& \)[/tex] is defined for any two integers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] as follows:
[tex]\[
a \& b = (a \times a) - (b \times b)
\][/tex]
2. Select sample values:
We'll use [tex]\( a = 5 \)[/tex] and [tex]\( b = 3 \)[/tex] as our sample values to check if [tex]\( a \& b = b \& a \)[/tex].
3. Calculate [tex]\( a \& b \)[/tex]:
First, we need to find [tex]\( a \& b \)[/tex] with [tex]\( a = 5 \)[/tex] and [tex]\( b = 3 \)[/tex].
[tex]\[
a \& b = (5 \times 5) - (3 \times 3)
\][/tex]
[tex]\[
= 25 - 9
\][/tex]
[tex]\[
= 16
\][/tex]
4. Calculate [tex]\( b \& a \)[/tex]:
Next, we need to find [tex]\( b \& a \)[/tex] with [tex]\( b = 3 \)[/tex] and [tex]\( a = 5 \)[/tex].
[tex]\[
b \& a = (3 \times 3) - (5 \times 5)
\][/tex]
[tex]\[
= 9 - 25
\][/tex]
[tex]\[
= -16
\][/tex]
5. Compare the results:
From the calculations, we have found:
[tex]\[
a \& b = 16
\][/tex]
[tex]\[
b \& a = -16
\][/tex]
Clearly, [tex]\( a \& b \neq b \& a \)[/tex] because 16 is not equal to -16.
6. Conclusion:
For the given sample values of [tex]\( a = 5 \)[/tex] and [tex]\( b = 3 \)[/tex], the operation [tex]\( a \& b \)[/tex] does not equal [tex]\( b \& a \)[/tex]. Hence, [tex]\( a \& b \neq b \& a \)[/tex] for these values.
The final result confirms that with [tex]\( a = 5 \)[/tex] and [tex]\( b = 3 \)[/tex]:
[tex]\[
a \& b = 16, \quad b \& a = -16, \quad \text{and} \quad 16 \neq -16
\][/tex]
Therefore, [tex]\( a \& b \neq b \& a \)[/tex].
