To identify which option demonstrates the associative property of addition, let's briefly define the associative property of addition. It states that the way in which numbers are grouped does not affect their sum. In mathematical terms, for any numbers \(a\), \(b\), and \(c\):
\[
(a + b) + c = a + (b + c)
\]
Now, let's analyze the options one by one:
A. \(m \cdot (x \cdot y) = (m \cdot x) \cdot y\)
- This demonstrates the associative property of multiplication, not addition.
B. \((x + y) + z + r = x + (y + z) + r\)
- This correctly demonstrates the associative property of addition.
C. \(y - (x + 1) = (y - x + 1\)
- This does not correctly demonstrate the associative property of addition. It appears to represent a different operation.
D. \(x \cdot (y + z) = (x \cdot y) + z\)
- This incorrectly represents the associative property and is not an accurate statement for multiplication or addition.
The correct option that demonstrates the associative property of addition is:
B. ((x + y) + z + r = x + (y + z) + r