. Prove that -x - y = -y – x; for all integers x and y.
-X - Y = - Y - X,
-X + (-Y) = -Y + (-X)..
Commutative Property: The order of addition does not matter.
a + b = b + a.
To prove that -x - y = -y - x for all integers x and y, we need to show that both sides of the equation are equal for any given values of x and y.
Let's start by evaluating the left-hand side (-x - y):
-x - y
According to the properties of addition and multiplication, we can rewrite this expression as:
(-1) * x + (-1) * y
Now, let's evaluate the right-hand side (-y - x):
-y - x
Similar to the left-hand side, we can rewrite this expression as:
(-1) * y + (-1) * x
Now, let's compare the two expressions we derived for the left-hand side and right-hand side:
(-1) * x + (-1) * y = (-1) * y + (-1) * x
Since multiplication by -1 is commutative (meaning the order of the factors doesn't matter), we can rearrange the terms on both sides:
(-1) * x + (-1) * y = (-1) * x + (-1) * y
Now, we can see that both sides of the equation are identical. Therefore, we have proved that -x - y = -y - x for all integers x and y.